Button quail CAN be tamed, but I think they only become truly tame (as in 'sleep on your lap' and 'come when you call their name') if they have been raised by humans with very much human contact - and possibly without other chicks around. They will need to see the human as their parent and later they might bond with their owner. This also means they will suffer when left alone.
Coturnix are probably the easiest to tame, based on what I've read - never had them.
Adults.. Well.. I'd say you can get them used to your presence, but they probably won't enjoy being held. Coturnix might accept it, buttons probably won't.
Bugs are a favorite of quail. Ants apparently taste bad, but they love many other bugs. People give them meal worms, but the fat content of those is supposedly very high, so it should be limited to a couple pr. bird pr. day.
I don't exactly tame my quail (buttons), but some of them will eat bugs from my hand - I could probably get all of them to do it if I had enough bugs and patience.