Each type of quail has a different incubation cycle. Coturnix quail take about 17 days. Bobwhites about 23 days. Button quail about 16 days, Gambels quail about 23 or 24 days, and so on. :-)
Go online here and do a google search. Hovabator is a good bator. Brinsea and R-Com are good as well. Several others.
They can run you anywhere from $100 to hundreds of dollars. Depending on how large, the quality and whether it has egg turners.
Generally no, they do not go broody. Most quail have lost the instincts to brood and raise young.
However, this being said, it CAN happen and occasionally does, but rare. But don't hold your breath. LOL You are best to get an incubator. :-)