To add onto what George said, you shouldn't play Persona Q until you've played Persona 3 or 4. And really I recommend playing it after having played both. The game is pretty much made for fans of the two games and includes lots of references and stuff.
Also, the Persona 4 Arena games take place after Persona 4 and also feature Persona 3 characters, so its story has spoilers for both games.
EDIT: Also, not entirely sure if I would compare the games to Final Fantasy like that, as the games do share more than just common elements. Though the first three (Persona 2 being two games) games are completely unrelated to the games that follow, Persona 3 and 4 take place in the same universe. Still, it's not like you'll miss out on much if you start with Persona 4, as it takes place at a different place with completely different characters and the story is unrelated to that of Persona 3. But there are a couple of cameos of P3 characters, and there is one part of the game that's filled with P3 references.
It is not yet known if Persona 5 will take place in the Persona 3 and 4 world though. But it is safe to say that there's no need to play any of the other Persona games first before starting it.