A good and solid answer was given already:

Modern firearms and grenades appear on page 268 of the Dungeon Master's Guide.

Please upvote that answer if you find it useful, I'm just copying it to comply with the guideline that every answer should stand on it's own even if the others get changed or removed.

However, a paragraph like this has been in all similar editions of D&D (2e, 3.5e and now 5e). I have tried that in the past but that little paragraph actually opens up a whole lot of other connected questions. For example how is armor treated if firearms are supposed to pierce them without problem. A lot of stuff that we take for granted in todays world simply has no rules. Not even conversions. D&D has no rules for chasing each other on horse drawn carts, so there is no easy conversion for modern cars either. The bottom line is, while this is an easy answer, it does not make for a good game conversion, it only leaves you with hundreds of similar questions that não have answers in the DMG or any other official D&D book. You will probably not be satisfied with your game experience if you take this answer and start playing.

So let me challenge the frame of your question. You want to play a walking dead style campaign with modern weapons. What you have as a tool to do so right now is D&D. And you run into problems because D&D was never meant to do that. There is no modern setting in D&D, it aims very specifically at sword & sorcery, dragons and heroes. No guns, no electricity, no medicine, no internet, but instead you got guys casting spells left and right.

Your question is a bit like having a car, and wanting to have a good sailing trip on the lake with your buddy. That's great. But you run into problems because your car isn't suited for it and now you are asking how to raise a sail on your car because you haven't found the mast yet. Well, there is no mast. It's not meant to do that.

While I'm sure a lot of people will be eager to sell you a mast and a sail and something to waterproof your whole car, let's look outside of the question for a second. How about getting a boat?

Looking around this site for a system that supports a modern day zombie apocalypse, there are threads here, however, they tend to be opinionated and not a good fit for our format.

What would probably work best is to find your local store and just ask them. If it's a good store, they will present you some systems that fit your needs and leave you at a quiet table where you can browse all of them to decide.

To sum it up: get a system that suits your needs. There are plenty out there. Go grab one and have fun.

Modern firearms and grenades appear on page 268 of the Guia do Mestre.

There are two possible answers one could give here, since you didn't ask explicitly for the answer to comply with the RAW.

1. O frio, difícil, RAW (Rules As Written) answer

   Just as @Sebkha's answer states, on Pages 267–268 of the Dungeon Master's Guide there is an official treatment of Renaissance-era, Modern-era, and Futuristic-era weaponry. If all you would like the statistics provided for explosives and firearms, skip directly to the tables at the bottom of Page 268.

2. O player's-experience-driven responder

   My friend and I both recently started playing D&D 5e and we wanted to try out a Walking Dead apocalypse style campaign.

   I will assume that since you said this, your friend is truly excited to play with weaponry that is neither common nor native to the standard 5th edition D&D setting, Forgotten Realms.

   I'm the DM so it's only him playing

   In my opinion, this the perfect situation for implementing a new set of non-RAW-compliant rules governing new weaponry. All you have to do is make sure that your one player is happy with the new set of rules and you're ready to get your campaign rolling!

Ok... now, for my answer.

TL, DR

Long story short...

1. I made a prototype weapon by starting with the stats I found in the tables at the bottom of Page 268 of the DM's Guide and added the weapon's trademark features/abilities.

2. I did a lot of iterative play-testing/tweaking with my players. I took particular care to make sure that I incorporated all the feedback I could generate and my players' could generate when deciding what changes to fix problems we observed with the initial prototype weapon's design.

3. (Repeat Step #2 if glaring problems were observed) Once I had incorporated these fixes into the prototype weapon's design, I began the play-testing process anew.

This process continued until we felt our D&D version of the weapon agreed with the version of the weapon that we knew from Stargate TV show.

My Personal Experience With Handling New Weaponry in D&D

I am an active 5th edition D&D Dungeon Master and I run a campaign with 5 very creative, adventurous players. Many of us have seen the popular Canadian-American Sci-Fi TV show called Stargate: SG-1 (as well as it's successors). So, we had the brilliant idea of trying to intersect the Stargate universe and 5th edition D&D universe. I will spare you all of the minute details, but the general idea was that shortly after the beginning of a campaign set in the standard 5th edition D&D Forgotten Realms setting, the planet was invaded by a Goa'uld System Lord's Jaffa army. The Jaffa soldiers in this army wield alien weaponry in the form of a Ma'Tok staff (staff weapon) and a Zat'nik'tel... so these were the weapons that I had to incorporate into the set of weapons already specified in 5th edition D&D rulebooks.

How Do I Arrive at Weapon Statistics that Satisfy the Player(s)?

1. Finding Prototype Weapon Statistics

   You know what weapon you would like to incorporate into the rules - you can imagine it operating and you have some sense of what it can do and what it can't do. However, you don't have any weapon statistics just yet. The easiest way to remedy this problem is to começo with what is given to you on Page 268 of the Dungeon Master's Guide.

   In my case, I prototyped an Antimatter Rifle as my Ma'Tok Staff weapon.

2. Play-test your Prototype Weapon

   Now, your weapon has statistics. Still, they definitely aren't exactly the statistics you would like. But, you can never really know how over-powered, under-powered, or balanced a weapon's performance will be just by looking at statistics. Thankfully, it's not too hard to set up a simple encounter in D&D and you've got a player that's willing to play D&D with the cool new weaponry you're designing specifically for their campaign. So, now you only need to whip up an encounter and give some monsters, some NPCs, some PCs, or a mixture of all three your prototype weapon. Then, take notes while the encounter plays out and make sure to get feedback on the weapon's performance from your player(s).

   In my case, I imagined an encounter in which my players' characters, armed only with the usual medieval weaponry found in the 5th edition D&D Player's Handbook, were pitted against a similarly-sized band of Jaffa warriors, equipped with Ma'Tok staffs and Zat'nik'tel sidearms, hiding on an abandoned crash-landed Goa'uld spacecraft.

Now... what happened?

• “The play-test went smoothly and the prototype weapon performed just as I hoped it would. It seems to be a balanced piece of weaponry, ready for utilization in my campaign.”

  Before you go ahead and incorporate the new weaponry into your campaign, try adding new and exciting elements to the encounters you've been play-testing the weaponry with. If you (and your player(s)!!!) are convinced that even if those wild and crazy encounters that happen time-to-time in the D&D game-play experience, then the weaponry is ready.

  What is important here is that you and the player(s) are happy with the new weaponry and the rules governing its usage. If you ever run into any hiccups that are directly related to the operation of the new weaponry in the campaign, remember that developing the weapon was a collaborative effort between you (as the DM) and the player(s). Because of this, you should all be able to come up with a mutual, quick, and reasonable fix for the hiccup.

• “The play-test revealed some pretty big imbalances that are inherent in the design of the prototype weapon (for example, the weapon hit its target too often).”

  First things first, present all of the imbalances / problems with the prototype weapon as it stands to your player(s). Make sure that you both agree on all of your findings and then incorporate fixes for all of the imbalances / problems you agree upon. Next, make sure that you collect any player feedback and incorporate those fixes into the prototype weapon's design as well.

  In my case, I found that the Ma'Tok staff was too accurate from long-range; in the TV show, the Ma'Tok staff was meant more as a weapon of terror that would intimidate and subdue the enemy and it exhibited a fairly poor accuracy when fired at a distance from the target longer than more than a few dozen meters.

  Now that you've identified all of the glaring problems with your prototype weapon, you are ready to iterate this process and whip up another encounter that focuses on testing precisely the imbalances / problems that you all observed in the previous play-testing encounter. Don't worry about the iterative process taking up your whole week. In my experience, it only took three encounters to iron out a balanced and play-tested statistical sheet describing operation of both the Ma'Tok staff and the Zat'nik'tel that I could agree upon with all of my players.

Aviso Legal

Digging into the details could lead to me writing a several-page-long answer, so I've tried my best to keep it compact. If I've left out something that you would really like to know from my personal experience with this process let me know and I will include more of my personal experience into my answer.

I would echo @nvoigt's answer: D&D was simply never meant to do this. It can be illustrated by a simple example: realistically, modern guns have a high chance to kill or disable anyone they hit. However you approach replicating this makes a mockery of the D&D hitpoint system.

If you're set on using D&D as a system, then a possible alternative is to a "dark fantasy" zombie apocalypse. The wide variety of undead in the MM should make it potential more interesting than the modern version. Ban (or severely limit) access to wizard and priest powers and spells. Likewise for magic items. Leave your Rogue and Warrior archetypes trying to deal with injury and disease via the natural healing rules.

User Nitsua60 made the excellent suggestion of using the Gritty Realism rules from the Adventuring Options section of the DMG, p266-268. That section has a number of other variants that would add to this setting such as Fear and Horror, Healer's Kit Dependency, Slow Natural Healing plus, as already suggested, Explosives and Firearms.

There are precedents from other games which have made use of a similar "survival fantasy" setting. The board game Zombicide: Black Plague is perhaps the most direct. Other useful places to look for source material and mechanics include the survivalist elements of the D&D Dark Sun setting and the Medieval material for Vampire: the Masquerade.

It might not be exactly what you're looking for. But it would be D&D and it would certainly be an interesting (and possibly fairly short) campaign.

To expand on nvoigt's answer, D&D is all about special people to accomplish heroic deeds. A cleric doesn't need a morphine shot to cast restauração, a wizard doesn't need a rocket launcher to cast bola de fogo, and a fighter doesn't need an adrenaline needle to use his Action Surge. These are things they can do because they were destined to be heroes. By contrast, a zombie apocalypse campaign is all about ordinary people prevailing despite the odds being against them. There's limited food, limited water, limited ammo, and a seemingly unlimited supply of walkers trying to eat your face. Sure, D&D has limited resources, like Action Surges and spell slots, but all of these recharge after either a quick rest or a good night's sleep, since you're always supposed to have access to them. No amount of tinkering is going to make D&D fit in a post-apocalypse, every-rotting-rat-corpse-is-a-feast setting - not without significantly rewriting and re-balancing the game, and at that point it'd just be easier to find another system.

There are a number of issues with handling modern weapons in a D&D game. And a number of solutions.

You could treat magazine-based firearms as being similar to repeating crossbows, and they do roughly the same amount of damage as said crossbows but with more range (treat bursts of fire as a shot instead of tracking 30 shots individually). Meanwhile, "primitive," weapons like actual crossbows (wood and glue; a modern produced composite crossbow would do normal damage but have reduced range compared to a gun) do reduced damage. For example a hunting rifle might do 1d8, and a wooden crossbow might do 1d8-1 (random numbers). To represent the technology disparity. Handgun does light crossbow amounts, etc. Or you could simply make guns incredibly deadly (1d20 damage).

Of course this isn't perfect. The primary problem is that D&D is heroic fantasy. Consider a 1st level fighter (using d8 HD), and a 0th or 1st level commoner (using d8 HD). Odds are the fighter will have 6+ hp, and the commoner will have 4 or 5 hp. Now consider a 5th level fighter (5d8) with his roughly ~25 hp or so. Now imagine the commoner and the fighter are attacked by a 0th level mercenary each. The mercs hit, and do 6 (on a d6) damage to both. The commoner goes down immediately, and the fighter is wounded but ready to deliver retribution. Now say it was your 5th level fighter PC that got shot. They just potentially shrugged off 7.62mm without a vest. So while the first solution works for any heroic settings, if you want things to be "gritty," and "realistic," in a modern-day zombie survival rpg then you've got a pretty big problem. If you try to rectify this by making guns ridiculously deadly (2nd solution) then you're likely to kill or frustrate your PCs when they die in one hit at level four. (All of the above assumes you look at HP like meat-points.)

To summarize my mess of thoughts: You can re-skin everything to make it modern and have an out of place heroic feel in what most might consider a grim/dark setting, or make modern weapons deadly beyond D&D's usual bounds and end up frustrating your players, at least until they reach higher levels and begin laughing off 1d20 (or whatever) damage.

I hate to say it, but you may actually want to consider using a different system for what you have in mind. Something like All Flesh Must Be Eaten (built for zombies), or Unknown Armies (ignore the rules for magick) which can offer both a modern setting as well as handle the issues guns bring up. But depending on the system you may need to home-brew a few things.

The important thing to consider if you are to use D&D for such a setting is on how you interpret HP. D&D is intentionally sketchy about it but typically most players still think of HP damage as 'damage', 'taking blows/cuts'. Under such an interpretation it will be quite hard to get the feeling of Walking Dead.

If you instead interpret HP as something more like stamina/getting cornered/luck then it all becomes easier. HP is simply how long you can avoid getting killed. Under such an interpretation it is no longer necessary to compare damages of ancient melee weapons to the damage of modern weapons, just pick values that makes sense for the system (That is, make the lightest of weapons do 1d4 and heavy weapons do 2d10 and other weapons inbetween). Base this primarily on weapon quality and how much you sacrifice to use it, i.e. a gun anyone can use in one hand does less damage than a weapon only a trained person can use on a tripod. Same thing with armors - full riot gear equals full plate (does not matter how they compare to each other as the players won't find plate armor anyways)

To keep up with the horror of getting bitten (certain death in WD), I strongly suggest that you let zombies get a bite in on critical hits, or at least cause some kind of trip/grapple/corner situation where a bite is not far away. That keeps the zombies dangerous and random even if players get a lot of HP.

The d20 system, which is maintained by Wizards of the Coast, has been adapted into many different, yet compatible rule sets. This means that you can (usually) pull many different rules from many different sets into one cohesive game. For example, you can find many compatible rules and items in D20 Modern or D20 Future, mais D20 Apocalypse (a supplemental post-apocalyptic module).

For example, starting around page 95 of the first book I linked, you'll find machine guns, pistols, etc, and you can read further for additional weapons. The damage values are diretamente translatable to DnD 5e, so you can treat those books as additional supplements to your campaign. As the DM, you can choose which weapons are available, or make your own using the ones in these resources as a guideline. While the D20 Modern and D20 Future books are intended to be stand-alone, there's nothing preventing you from using the parts you want in DnD 5e.

When considering damage for any weapon, including modern day firearms consider their damage against a regular person, who in DnD tends to have 2-3 hit points.

Think about a glancing blow to the limbs, more than direct hits to the chest or head, which would be more akin to a critical hit.

A small, low-caliber handgun might be no more damaging than a dagger with non-critical hits, and could easily deal 1-4 damage.

Higher caliber handguns would act more like short swords and hand axes, and deal 1-6.

Shotguns and rifles could be 1-8 like battle axes and longswords, and so on.

So, that critical hits with a .45 against a regular person with 2-3 hps, can still deal an average of ~7 hp of damage, which would be a lethal level of damage for anyone not immediately rushed to hospital (i.e. negative hit points and draining fast.)

For rapid-fire weapons treat a hail of shots as a single attack, so that a sub-machine gun fires, say 10 bullets a turn, but that hail is a single attack of 1-8 dmg.

While some may balk at the unrealism of this comparison (guns are incredibly damaging to internal systems) we're looking at in-game effects across short term, combined with a desire for balanced game-play.

Remember that using a ranged weapon against an adjacent opponent with a melee weapon has some disadvantages, and you may find players pick hand weapons over guns more often that not.