I haven't done quail, but a good friend of mine has. They're beautiful little things. I think your plan sounds good. They do need to be kept enclosed, because they aren't like chickens who will come home to their coop every night. (At least the breed my friend had wasn't). Hardware cloth is your friend in terms of predator prevention for birds. The only thing with hardware cloth is that the spaces between the wires is smaller -- maybe a half inch -- so hopefully that is big enough to let their little poops fall through into the grass. I think it would be.
A hardware cloth 360 degree envelope is key. We have our entire chicken coop and run (16 x 20') covered on all sides, including the bottom and top, with hardware cloth. We live in the country (loads of predators) and in 9 years we've never had any predator get in. Cats, raccoons, and foxes (biggest danger) can't tear it with their teeth or claws, and can't dig in because the bottom is also covered. And the spaces are too small for the smaller predators like weasels and snakes to get in. You can buy hardware cloth at any DIY store and it's pretty easy to work with. You can bend it with your hands, cut it with a wire cutter, staple it to wood, and sew it together with wire -- whatever the situation calls for. Wear gloves while you're doing all that -- the cut ends are sharp.
In the areas where you staple the hardware cloth to wood, be sure to use longer, heavy-duty staples -- they're like an inch long. The regular staples that people use in staple guns are maybe half that long. Raccoons are very strong and they have hands. They can pull those shorter staples out by grabbing the hardware cloth and tugging hard.
I'm sure the local quail breeder can give you tons of advice. I think there's also info about quail on the Backyard Chickens website, you could check that out. Lots of info on there, including coop and mobile tractor ideas and plans. Have fun!