Ummm, I really don't want to stir the pot here so instead of going with the standard "Yes you can" or "No you can't" side of this debate I thought I would toss out a possibility that the "No you can't" people may not be thinking about.
Most of the quail raised today in captivity came from quail that were raised in captivity who came from quail raised in captivity <repeat until nauseous>. As a result, calling them wild gamebirds is rather over stating their origins a might bit. Heck, many of the strains of coturnix out there have never been seen in the wild (The A&M White for example). Somewhere, some place down the generational chain, many of these quail lines were brooded, housed, and allowed access to chickens. As a result, they have been exposed to the chicken yuckies <a good term to cover 99% of all chicken specific diseases>.  I'm thinking the guy back down in history that decided he was going to eat the next thing that popped out of a particular bird's rear and began the craze of raising chickens wasn't overly concerned about disease spread when he discovered that the smaller thing that popped out of a quail's rear tasted good when soaked in sour wine for a few days and began the craze of raising quail (as well as getting stinking drunk on pickled quail eggs and beer). 
In other words, just like any animal, resistances can develop over years of captivity. While wild quail may keel over at the mere mention of a chicken yuckie, domestic quail may laugh in it's face. To automatically presume that all quail will sicken and die if in contact with chickens is preposterous.  This would be especially true of those quail that are hybridized and have been created by man to start with, such as the jumbo and A&M white coturnix and the Butler and Georgia Giant bobs. A little research and common sense will show where these can be safely housed near or perhaps even with chickens.
Now, would I head out and grab ten wild quail and toss them into a chicken coop? No way in the world. Not only would I then be running the risk of giving the quail a chicken yuckie but I would run the risk of giving the chickens a quail yuckie.  Even with domestic quail, I would strive to keep them separate in some manner if for no other reason that anything smaller than a chicken looks like a grasshopper to an orpington and that chicken toes look a lot like worms to a quail. Will I brood chickens and quail together? You betcha. I did it just recently and did not lose a single bird to any disease. Are they still in the same living quarters? No... chicken toes look like worms.