Thanks you both for the responses!
I think that after the babies hatch I'll move mama quail and babies to a temporary cage, as well as the males, and then I'll make the tank that they live in chick friendly. They have a fairly deep water dish, so I'll get some pebbles in order for the the little ones not to drown.
I'll keep a careful eye out for any aggression that the males may display towards the chicks, and I'll separate if need be.
My mama quail and her brother were actually hatched by their quail mother, so I guess she might have passed the instinct to brood onto her chickies.
So I figure I'll put the events leading up to her going broody:
I was going to feed them and I saw this tiny little egg that had been laid right near the food dish. I showed my little sister, who accidentally dropped it on the ground(in true little sister fashion) so she ate it for lunch, and that was that, or so I thought.
The next day it was time to clean out the tank, and I went to remove the small plastic 'den' on the second story of their tank, only to find an egg! I decided to leave it, and what do you know, six days later there were six eggs in the nest (she skipped laying for a day)
She then decided that sitting on them would be a good idea, and here we are!
If the tank layout could be changed to be suitable for baby quail, that would probably be best. What I do in my cages is: removing all hiding places (because weak babies seem to go for any large, darkish object they see when they are cold - and if this object isn't mom they might end up dead) and anything they could get stuck in or behind (sand bath for instance), placing the water fountain either on the bedding or on a piece of cardboard on the bedding (usually it's on a brick to keep bedding out of the water - this is removed), replacing the feed dish with a very shallow one so the babies can get into it and possibly grinding the feed.
Depending on what you use for water now, you might need to replace it with something chick-friendly and perhaps fill it with marbles or pebbles so they can't get wet.
Once the chicks are a week old, the cage content can usually be placed back in - by then they are so strong they can find their mother when they need her.
Now, I've never tried hatching chicks in a cage with two males, so I have no clue how they might respond to this. Perhaps both males will try to help raising the chicks. If they allow each other to and the hen allows them to, let them. If either male won't let the chicks sit under him, get him out of there - same issue as the hiding places that need to be removed, even if he isn't hurting them they could die because they try to follow him and gain no warmth by doing so.
If you think it's better to move mom and brood, that can probably be done as well. If her partner sits with her and the chicks after they hatch, I'd move him as well, but she should do just fine on her own.
With regards to removing cagemates when a hen is broody, in my experience it actually doesn't bother them. Not even if it's the mate. But some hens might not like it.
Thanks for the detailed response!
How exciting! Hopefully your hen will do a great job. Often the male becomes very interested in helping right on hatch time so look out for that.
I'm restricting how much light mine get at the moment as we are hopefully moving soon and I don't need babies! That hasn't stopped one of them though and her sister (who is very broody) was already trying to sit on the few she'd accumulated. I'll just have to keep removing them.
Let us know how your hen goes.
Moving house is exciting (and stressful, lol) having persistent broody button quail probably isn't going to help;
How are you going to move all of your buttons! Also, slightly unrelated, but what was the outcome of that oddball quail that you had? I think your daughter called him/her lynx?