Who cooks: Q. Sort of. He doesn’t actually cook he just magics the food into existence.
Who does the laundry and other chores: Q, but again only sort of. He’s too lazy to do them properly.
How many children do they have:….hmmmm just one. It took Q /forever/ to convince Picard. Q is surprisingly good with children and Picard is kind of relieved because he’s not entirely sure what to do when it starts crying. ( also it’s probably not a human baby)
Who’s more dominate: That depends on whether or not Q has been annoying to the point that he needs to be punished [:
Favorite nonsexual activity: dragging Picard off to do random things instead of run his ship
Their favorite place to be together: Picard prefers the Enterprise, Q is happy wherever Picard is
Any traditions: if Picard refusing Q’s advances and then grudgingly giving in counts as a tradition?
Their “song”: “Shut Up and Kiss Me” Orianthi
What they do for each other on holidays: Q gives Picard loads of things he doesn’t need and probably can’t use(Picard has a closet full of useless things Q has given him because despite himself he can’t throw them out) Picard is slightly less grumpy and gives Q small earth things
Where did they go for their honeymoon: France
Where did they first meet: all that stuff that happened in encounter at farpoint I think
Any pets: Q frequently brings home alien pets but Picard refuses to let him keep any of them
What do they fight over: anything and everything but especially Q stealing all the blankets and popping into places unannounced
Do they go on vacations, if so where: yes Q drags him away every chance he gets to various planets around the galaxy he calls it Picard’s “education” about how insignificant humans are but he actually just likes to see Picard experience new things