Chicks and quail chicks should be fine together for a week or so, at least until the chicks start becoming larger than the quail. Leave the chicks in the incubator until they are 3/4 to completely dry and fluffy; then transfer them to a brooder.
You should prepare the brooder today. Line the bottom of the brooder with wood shavings or chopped straw. Don't use cedar shavings, as cedar irritates chicken respiratory tracts. Place a chick waterer and feeder (or two) in the brooder. You might want to put some small stones or marbles into the water to make sure that the tiny quail chicks don't drown in it. And I assume that you have some sort of heat source, such as a heat lamp or special chick brooder, to put in the brooder. I also like to put a thermometer in the brooder in order to tell how hot it its.
The temperature during the first week should be 90-95 degrees F., and then drop by 5 degrees each week thereafter. Chicks that are comfortably warm will scurry about the brooder, eating, drinking, and sleeping. Cold chicks cluster together for heat, and hot chicks spread to the outside of the brooder, panting.
For the first day or two after the chicks hatch, I usually cover the shavings/straw in the brooder with paper towels. This ensures that the chicks learn how to eat real feed, and not just stuff themselves on wood shavings. Of course, they soon make a mess of the paper towels, and when that happens, its usually time to remove the paper towels.
When you put the chicks/quail into the brooder, dip each bird's beak into the water. This will show them where the water is, and teach them (hopefully) how to drink. You could also try putting each chick near the feed so that they can know where that is, but most chicks don't start eating until they are at least a day old. To give the chicks extra energy and nutrients, I usually put probiotics and vitamins/electrolytes in the chicks water.
And thats about all. Just monitor the chicks for any signs of a disease, such as Coccidiiosis, and they should be fine.
