On a dark colored button quail, the males will always have rust colored feathers around the vent and the females won't, so it should be very easy to tell whether Luna is female.
From your description, I don't think they are going to sit on the eggs. First of all, you haven't had them for very long, secondly the male is chasing one of the females meaning there is a certain amount of stress involved. And then there is the enclosure, which doesn't really have any good nesting sites - the hut in the far left corner might work if the birds would enter it and lay their eggs in there, but I have tried something similar and mine didn't so I'm guessing yours won't either. Also, the eggs are out in the open and being moved around, rather than in a nest - some people say button quail will lay their eggs all over, then decide on a place to nest and gather all the eggs in this nest, but mine never have. They will only incubate the eggs that are laid in the nest or possibly gather eggs within 10 inches or so from the nest.
My buttons seem to prefer making their nest behind spruce branches, so if you have any of these, you can try placing some in the two far corners, trying to make it as dark as possible there. Change the branches as they loose their needles, unless there is a nest behind them - then I wouldn't disturb them, unless the nest has 12+ eggs and there is still no broody hen. With 12+ eggs, you can remove them all, place a new branch if needed and let them start on a new clutch and hope they'll go broody.
If they are not using the hut (apart from parching on it), I'd remove it, it's taking up a lot of room.
