Idk about putting them in the quail enclosure as I don't have quail, however, I did breed and raise them for my lizards years ago.. a 3 tier plastic drawer bin.. top drawer for adult beetles, middle drawer for newly hatched mealies, and bottom drawer for larger mealies.
As you use the larger mealies the second drawer will be dumped into bottom drawer, top into Second, beetles will die and need to take some larger worms to put in top to pupae into beetles to lay more eggs.. sounds way more complicated then it is lol I did it for quite some time.. I know supers are larger with larger Beatles so a deeper drawer method might be better for them..
I had them on an oatmeal substrate and fed them lots of different veggies for water content etc.. they also make water gel things. Anyway, good luck!