Fresh quails eggs are generally not stored for extended periods. At a cool room temperature, you can safely keep them for three to four weeks. Is is advisable not to wash quails eggs prior to storing them, as the protective layer (called the "bloom") will be washed off during this process.
The bloom protects the contents of the egg from bacteria entering the porous shell. Quails eggs from a properly managed farm, will generally be clean and collected shortly after they were laid. A bit of dirt on the eggs is nothing to be worried about. If you are planning on frying the eggs, or using them raw, a gently rinse and rub under running water that is slightly warmer than the egg will remove all dirt and prevent contamination.
Eggs that have been stored for slightly longer than a week tend to be easier to peel. This is due to a slight moisture loss in the contents of the egg.