Well they don't eat layin mash. They need a feed formulated for game birds. Mine get game bird starter all of their lives. Feeding chicken feed isn't very good, they need more protein plus some other things.
I'll just tell you what I feed my coturnix from hatch to dispatch.
Purina Mills Game bird starter. 30% protein, product #5419.
They also make a breeder formula: Product #5438, but switching up coturnix feed, may have undesired results.
No matter what you feed them, keep it consistent. If they need extra calcium, then feed appropriately formatted oyster shell, but a coturnix hen will lay fine on any Game bird starter!