De KnowYourMeme.com , ênfase minha:
One of the most popular facepalm images is one of Captain Jean-Luc Picard from Star Trek the Next Generation taken from the episode “Deja Q,” in which Picard is shown reacting to a stressful situation
(Este foi o primeiro resultado do googling Picard facepalm btw.)
De Wikipédia :
"Deja Q" is the 13th episode of the third season of the American science fiction television series Star Trek: The Next Generation...
In this episode, as the crew of the starship Enterprise struggle to prevent a moon from falling out of orbit, their situation is further complicated by a visit from a powerful nemesis named "Q" (John de Lancie), who informs them that he has been stripped of all his powers and must live out a mortal life.
Aqui está um vídeo:
Nesta cena, Q (um personagem que o resto da equipe acha muito chato) está explicando para a tripulação da ponte que ele foi despojado de seus poderes e é humano agora. Picard está reagindo a essa notícia com ceticismo e aborrecimento, baseado em encontros anteriores com Q.