Everything everyone has said is good advice. Just as a side note because I am a nerd, moulting is based on several factors, primarily food supply and the photo period. Naturally birds moult over the summer months, because food is easier to come by, this is triggered by the photo period. (You can artificially trigger this with lights, give them 12+ hours) This also explains why most parrots, and indoor birds moult year round (ample feed and long photo period).
Birds a primarily sight based creatures, so overloading that sense (24/7 light for example) will stress them out. The next big ones are sound and sudden temperature changes. If you can control these factors you can do a lot to ease the stress on your birds. Provide ample and healthy food, along with a good day/night cycle, and keep the temperature constant, and the area free from loud and sudden noises, and they will recover. One way to help is to cover part of the cage with burlap or some other partially see though material, this gives "cover" for them to hide in, and can help to relax them.
Another thing to look out for are injuries on the birds. Check the bare spots especially as quail skin is thin, and can easily be rubbed raw. Any birds with open wounds should be treated, this will help reduce stress and save lives.
Also to echo what was mentioned previous, make sure they are getting clean water and good food. You want them on 20%-28% protein. I personally use 24% or 28% depending on what the bird is going to be for. Make sure their water supply is clean. Dirty water can lead to stress and illness, as can poor food quality.
That is all I have.