When it comes to Game Birds (by state definition, quail are game birds) the G&F sets the regulations. If you want to legally keep quail in Wyoming, they will be kept in an approved cage. The local game warden will inspect your set up prior to a permit to raise quail being issued.Quail are wonderful animals and don’t deserve to live in a cage. Please don’t do that.
The Cheyenne animal rules are here:
City codes
If you can’t have chickens, I suggest you buy eggs from someone local. There are a few of us that advertise on Craigslist at very reasonable prices. It’s not as fun as having your own chickens but you can still get colorful eggs.