A way to tell if a quail egg is fertilised and if a chick is growing inside is to candle the egg.
Although candling a quail egg can be difficult due to the dark markings on the egg shell.
Candling is a way of checking the fertility of an egg and the development of the embryo, with the use of a light source in a darkened room. In a darkened room, carefully hold the egg up to the light to observe the contents of the egg.
Click on this link for more information about candling an egg to see if the egg is fertilised.