Predators of a variety of types can be a threat to quail. Avian predators are a threat to adult birds while nest predators, such a racoons, opossums, armadillos, and bobcats are also very harmful. The path for a bird to hatch and then make it to adulthood is a perilous one.
No doubt some quail managers put a lot of effort into predator control, and I've done some of that, too, but as we know, predators have been around for as long as quail have, so there must be something else that's in play here. In the opinion of most, the change that has been been introduced that is most negatively impactful on quail populations is simply the loss of habitat. You're not going to see quail in cities, in monogamous pine tree plantings, in large irrigated row crop fields, in bahia grass planted for cattle, in timber tracts with a high basal count with little ground cover, or smaller isolated tracts that cannot handle the weather variations needed to sustain a population.
Wisely managed quail properties have 3 key components, and when you eliminate one or more of them, things don't' work. You need nesting cover, you need food, and you need escape cover. While predator control can be beneficial if done on a large and consistent basis, in many cases you simply open up the avenue for expansion of a different species when you remove a competitor. Most people do not have the time or resources to do this on the scale needed. Hence, the answer that I think is a better management target and solution is to simply have the needed cover in place where predators can't so easily have a free lunch.
There's a reason that pen-raised quail only live for a few days at most after being released and that wild birds can continue to flourish on a wisely managed tract. One is wary and one is not, and in addition, one knows where to hide while the other stands out in the open and says, "What is that fast creature flying up overhead?"
Predator control is nice, but habitat management is better.