Well, the story of each Persona is different so you won't miss nothing if it is your first game (just a few details about Igor that appeared in both games).
In the case of Persona Q, it features its own story based in the characters of both Persona 3 and 4, but it isn't necessarily needed to understand the story (unlike the fighting spin-off which name I don't remember haha)
So enjoy Persona 5 now if you have a Japanese copy or in February once it is out in the rest of the world.
If you have a PS Vita (or PlayStation TV, this thing is 40 bucks) I deeply recommend playing Persona 4 Golden and Persona 3 FES.
Despite that I sold my vita I'm buying a PS TV so I can play the rest of the games I bough, including these two.