Sky, correct in that it is generally recommended against mixing quail and chickens because of the diseases that chickens carry, even if not manifested, and that can then be fatally passed to the quail. That being said, in all of my reading in BYC, those who do want to hatch quail naturally have reported successful hatching under bantams. I don't know of their health successes after hatching, however, especially as people generally don't like to report failure when they go against the overwhelming recommendation...
Hey guys! Sorry it took forever to get back to this thread. In the spirit of being honest so other people can use the information, I've come to report both success and failure.
Out of our 25 jumbo Wisconsin bobwhite quail eggs 15 hatched. Unfortunately, the night I found them hatching I discovered it by three little dead ones out of the nest. I don't know if the bantams pecked them to death, squashed them, or if it was because three hens sharing the same nest made the girls clumsy.
Immediately I removed all the chicks and eggs to the incubator inside. One more chick died in the incubator, visibly weaker than the others. I think it got pulled from its egg too early while sitting under the chickens (another reason I say the three sharing the nest must have made them clumsy). I was certainly heartbroken and felt guilty by the deaths of the four, since I knew simply incubating them inside could have prevented the whole thing. The 11 others hatched healthy though.
Determined not to have any other mortalities and knowing many quail chicks die in the first few days from starvation/dehydration, I spent many hours teaching them individually where the food and water was. Dripping water from a syringe into their water dish works like a charm btw! The little chicks dart over to "eat" the wiggly droplets. From there you ween them to drink from the dish the water drips into. After 3-5 days, they were actively eating/drinking by themselves.
I started giving them minced garlic in their water when they were about 2 weeks old to prepare for moving outside with the chicken (only one remained after one was found dead in the yard and the other disappeared). They went outside around 3 weeks (near fully feathered and able to fly well already). I kept then on garlic water for several months. They now drink regular water and get along with the bantam just fine.
We did loose one quail to a mysterious illness(?). One day she was fine, the next couple days she was all puffed out and weak (couldn't hop on even a small log while the others could easily leap to the tallest), and the next day we found her dead. I'd started giving them garlic water again on the second day of her being sick, and though she didn't make it none of the others ever got sick. I really don't know what killed her. It certainly could've been something picked up from the chicken, or something she ate, or something else all together. I just don't know.
Our 10 remaining quail are all beautiful adults now. They coo and sing and talk to you, I love them

Theyre about 7 months old, still live with the bantam hen, I haven't given them garlic water since a couple months after the one adolescent died, and the rest all appear healthy. My conclusions come to this: never let 3 bantams hatch a single nest of fragile bobwhite chicks. Secondly, yes quail can eat garlic, but there's not enough evidence in my experience to determine if it prevents illness.
Just because of that experience I doubt I'll ever hatch quail under bantams again, though not saying a SINGLE bantam couldn't do the job (as many people have seen before with various success). From now on I'm probably gonna incubate indoors or stick eggs under a broody bobwhite hen (if I can get them to go broody of course). Hope this helps anyone looking for info on the topic. Good luck!