Mountain Quail
Even though I have never seen these, I am pretty sure that the Mountain Quail would need more room than your common Jap.
When you say Coturnix quail, I assume you mean Coturnix japonica. There are a total of 8 different species of Coturnix Quail, and they are very very different. I, myself, breed 4 different Coturnix species. One of which is the Coturnix chinesis, which I am guessing you mean to be the 'button', even though that isn't a good name for them, as the proper 'ButtonQuail' is very very rare, fetching hundreds of dollars per bird, while being completely unrelated to what you would be thinking of. There are 11 species of ButtonQuail (Turnix).
Coturnix japonica (or simply Jap Quail or Jumbo Quail), would only need a small hutch. Even the small triangle Guinea Pig hutches would fit 5-6 easily (depending on size, of course).
Coturnix chinesis would also fit a pair or trio into the same sized hutch.
A proper ButtonQuail would need a lot more room, probably a good sized aviary.
Button Quail (Barred)
Coturnix chinesis (Male). More commonly referred to as the King Quail, and for some reason in America the 'button' Quail, sometimes (even though they are not button quails, and far from it).