Some hens seem predisposed to going broody. I’ve never had much luck, just a couple over the years. It was always hens who had a nice secure little cave nest. If you want to encourage her, make a really nice place for her to hide and nest.
Button quail are more likely to go broody than coturnix. If you give them plenty of space and plenty of places to hide, they might surprise you. I know someone who went on vacation and the person who was watching their birds didn't take out the eggs. She came home to chicks.
I've had several hens go broody countless times. Leave their eggs for them and give them places to hide. They also like using straw to build nests, so perhaps provide straw. Also, if you don't already, consider keeping them in pairs. Give them plenty of food and water too.