I have some that will and some that won't. Originally, I thought I'd be able to house the buttons with the Coturnix, so I introduced them in small cages inside the big cage, and then after about a week, tried them out. Two of the pairs had no issue, the other two button pairs were chased and pecked at by the Coturnix. I tried again a couple weeks later, same deal. I swapped the pairs and tried again, still no luck, so I just got more cages for the buttons. I wound up with an extra Coturnix roo and an extra Button roo, both of whom were mouth breathing. I put them in the same hospital tank and they get along like they were made for each other. The Button will sleep snuggled under the Coturnix, which helps keep him warm. They're slowly recovering from whatever bug they had and none of the others seem to have caught it. However, I don't think I can separate them after this, so I'll probably get them a bachelor cage.
The Buttons and Coturnix that live together pretty much ignore each other unless mealworms are involved. Then they'll spat at each other and fight over them, but there's never any real aggression.
-Spooky