The basic procedure is similar to washing someone's hair in a sink. Prepare a washing up bowl of nice warm, but not hot, water. Add some soap/washing-up liquid or shampoo. It is important to hold the bird firmly but safely to prevent any attempts at escape. The best method is to hold your quail by the legs with your index finger between the legs and your thumb and forefinger gripping the legs.
You must now rinse your quail with plenty of clean water. You will probably find that you need a couple of bowls of water to get rid of all of the soap.
Unless you are a magician and can get your quail to obediently sit in front of a radiator, towels and a hairdryer are your best bet. Again dry the feathers as if drying your hair. Rub carefully with a towel to get the worst of the wetness out without damaging the feathers. To get rid of the last of the moisture, use a hairdryer as if drying your own hair. Don't let the top feathers get too hot as they could get damaged.