Quando são as estações do calendário Forgotten Realms?

The solstices and equinoxes changed recently (1488 DR)

At the end of the Second Sundering, the traditional days of season changing shifted. The Guia do Aventureiro da Costa da Espada has the details (pg. 15):

The worlds of Abeir and Toril drifted apart in 1487 and 1488 DR. In some places this change was accompanied by cataclysm, while in others the shift went without notice. Astronomers and navigators who closely watched the stars couldn't fail to see that there were nights when they seemed to hang in the sky. The winter of 1487–1488 lasted longer than normal. It was then noted that the solstices and equinoxes had somehow shifted, beginning with the spring equinox falling on Greengrass of 1488 DR. The seasons followed suit, with each starting later and ending later.

This shift in seasons has caused some sages, and the priests of Chauntea, to consider changing the marking of some of the annual feast days, but most folk counsel patience, believing that the seasons will fall back to their previous cycle over the coming years.

Assuming the spring equinox is now in the Greengrass holiday, you can assume the spring lasts until Midsummer, where the summer now begins. Somewhere near the end of Marpenoth marks the start of the autumn, which lasts until Midwinter, where the winter now starts.

Old calendar

If you, as the DM, want to use the old equinoxes and solstices to mark the seasons changing, you can look at the Configuração de campanha de reinos esquecidos from the 3.X edition of D&D, which includes the following image:

You can find detailed information about the calendar of Harptos on the FR wiki, which even has a nice picture depicting when the seasons used to start for all the past editions of D&D:

• Winter Solstice (Nightal 20) is the beginning of winter.
• Spring Equinox (Ches 19) is the beginning of spring.
• Summer Solstice (Kythorn 20) is the beginning of summer.
• Autumn Equinox (Eleint 21) is the beginning of fall.

Yet in the 5e, following the Second Sundering, about a 40-day shift has happened, as mentioned in the answer by Kuerten. Whether this shift will be permanent or not remains to be seen. A Guia de Estilo de Reinos Esquecidos, distributed by Wizards of the Coast to official play writers, is available on the DM's Guild (a copy of which you can download for free aqui), e isso aligns the month of Hammer with Earth's December, while in all the earlier-edition materials Hammer was January. By the way, it is also worth noting that there has been some controversy on this issue as described in the Candlekeep forums in the beginning of this year (2019).

Regarding the climate in Waterdeep, details can be found in the AD&D 2e boxed set The North: Guide to the Savage Frontier. In the book O deserto, Waterdeep is described as the roughly the latitude where the subartic climate ends, yet the warm ocean current keeps the weather warmer for the coast:

From the Spine to Waterdeep, subarctic weather is found. The coastal areas as far north as Port Llast are temperate climates, due to the warm ocean currents running along the coast.

The majority of the North, including the islands and the Spine of the World, fall in a subarctic clime. Long, bitter winters typically last from mid Marpenoth to late Tarsakh, with temperatures dropping to -30° F and rarely climbing past 40° F.

True temperate climate begins south of Waterdeep, yet the coast between the City of Splendors and Port Llast stays warmer and wetter throughout the year than inland regions. ... Summers are warmer here, with temperatures reaching 100° F or higher, but are relieved by nearly constant sea breezes.

Please note that the data in these sentences are based on the old version of the calendar, so apply a 40 day shift as you see fit.