Chickens have been domesticated and carried all over the world for long enough that they have immunities to, and often carry, diseases that many game birds have never encountered. There are a lot of things that don't affect chickens more than a cold affects us, if that, because of their immunity, but can cause serious harm to birds with no immunity.
However, if you get coturnix quail that are hardy against those diseases, or if your chickens aren't carrying those diseases, it's fine.
It's kind of a shot in the dark as to how it'll work out, keeping them near each other. They shouldn't be kept together because of behavioral and size differences, though.