I only have buttons, but as far as I know, most of it applies to coturnix as well:
Usually they hatch in the evening or during the first part of the night - at least I assume so, as they are usually dry when I first see them and that is nearly always in the morning. Mommy might have stayed on the nest during the hatch even if it happened during the day - I don't know, but for all I know she stays there when it happens at night. Or leaves and finds her way back to the nest in the dark without me knowing - strictly speaking that could happen, I don't have surveillance on her. But I assume she stays put.
Occasionally I've had late hatches that were not dry in the morning when the rest were ready to leave the nest. I've found those dead in the nests later when I went to remove shells, unhatched eggs and such. It seems the mother decides to leave the nest to care for the ones that are ready to start feeding even though she can probably hear the distress calls from the wet chick.
I don't think she actually removes the shells, but most of them tend to be laying around the nest in the morning when the chicks have hatched. Maybe it's more like she's not gathering them under her anymore when they get displaced from her moving on the nest.
She might return to the nest even after all chicks have left it for the first time, but it will be a random occurrence - she'll most likely just sit in the different corners of the cage or near food and water. Don't underestimate quail chicks though. I find it highly likely that they can at least get out of that nest box. Getting in might be stretching it, but you never know. Actually the nest box might pose a danger as a chick could get out of it before the rest are ready to leave, and be unable to get back in.
Beware of corners they could get lost in - if they can't find their mother, they can go cold and die very quickly during the first few days.