I strongly doubt there are no birds except for the hummingbirds that lay smaller eggs than buttons - I haven't exactly made side-by-side or weight comparisons but I think my budgies lay smaller eggs and there must be lots of finches and such that do as well? I certainly can't imagine my tiny zebra finches laying a button quail sized egg

I think the internet says button quail weigh around 50 grams - and zebra finches around 16.
I'd make the incubation time 16-18 days - it might be 16 in an incubator, but my buttons hatch their own eggs and I don't think they've ever done this in less than 18 days.
Other facts to include:
-They love dust baths
-They like small bugs like meal worms and can often learn to eat them from your hand - but some suggest if you feed them meal worms they might mistake the toes of other birds (chicks in particular) for worms and peck at toes. (I can't confirm this as I don't feed my birds meal worms from fear they won't be able to raise their own chicks - but I have had two hens with an interest in chick toes, and those could have been fed meal worms before I got them). I have fed a small cricket to my birds once, they really liked that and there is no chance of confusing those for toes. They don't seem to like ants much - they ate them the first couple of times I offered them and ignored them the third time. Maybe they taste bad - at least the kind I had in my garden. Ant pupae are supposed to be a much loved treat, but I haven't tried those.
-If you offer them seed mixtures like budgie seed, they will eat this to the exclusion of all else (except bugs) - this makes it very hard to balance their diets so they get enough protein (at least 19%), calcium and vitamins. So only offer seed as an occasional treat, game bird starter crumble is much better as their stable diet - and make sure to give them crushed oyster shell (it has to be pretty small for them to be able to eat it) free choice in a separate dish so they can eat what they need for calcium. Too little protein can cause feather picking and too little calcium and vitamin D3 can cause egg binding. And of cause a malnourished bird doesn't live as long as a bird on a healthy diet.
-You can use many different kinds of bedding, but they have different drawbacks and advantages. Sand for instance turns the entire cage into a gigant dust bath, which the birds love, and you can clean it with a strainer (no need to throw it all out when you clean the cage) but depending on temperature, humidity, number of birds and cage size, sand can promote the forming of 'poo balls' - the birds step in still wet droppings and the droppings stick to their toes and dries into a hard ball that can become so big the birds have trouble walking or the ball can restrict blod flow to the toe so it dies and falls off. Shavings make the droppings dry faster and also allow them to be buried in the bedding, so the birds don't step in them, making poo balls way less likely to occur. Wood chips dry the poo, but doesn't allow it to be buried. Straw lets it be buried but doesn't dry it as much and it can be hard for chicks to get around in straw.
-If poo balls do occur, the best way I've found to get rid of them is to get a small cage with a bottom that will hold water, place small pebbles in the cage till they cover the bottom, put the bird in and then pour warm water into the cage till it just covers the pebbles. The water will dissolve the poo ball and the pebbles will scrape the dissolved matter off when the birds walk on them. Sometimes the poo ball is gone in 15 minutes, sometimes it takes 1½ hours. Just leave the bird there till it's gone. Some people remove poo balls by holding the bird, dipping the foot in water and scraping, pulling, cutting and all kinds of things at the poo ball, but there is a risk of hurting the bird this way - you could pull the toe off, and the handling stresses the bird.
-Button sounds: A low, straining "squaaa-aak... squaaaa-aak... squaaa-aak" tends to mean your hen is laying an egg and is pushing hard to get it out. A loud "PEEEEW-peeew-peew" my mother translates into "DOOOON'T like this!" - sometimes it's more like "PEE-EEW-peew-peew-peew-peew" (or "IIIIIII-do-not-like-this"). This is the sound they will make when calling for their friends - or when they've been in the aforementioned poo ball foot bath for an hour or so an thinks it's about time to go back to their cage ^^. They also have a sound for "Stay away from me!", which they use when another button is chasing them or gets too close to their nests (I think those are the same sounds, but I'm not entirely sure) - some don't mind at all when others get near their nests and others will attack anyone that's within 1½ feet of the nest. This sound is best described as 'aggressive chirping'. And then there's the tid-bit-sound, "cluck-cluck-cluck-cluck-cluck" which the roo will use to call the hen when he's found something particularly delightful like a bug or seeds, and both roo and hen will use it to call the chicks. And of cause their usual small, cute sounds that don't seem to serve any real purpose other than letting each other know they are alive. There are more sounds than that, but those are the ones I usually hear.
-They like to have places to hide in their enclosure, but don't seem to appreciate actual houses. They'd much rather have spruce branches, fake plants, tunnels, hay, platforms (they seem to like hiding under them more than actually standing on them) and such.
-If you want your birds to brood their own young, give them as big an enclosure as possible and make some private corners in this enclosure. This can be done with spruce branches or by placing a 10x10 inch platform maybe 5 inches above the substrate in the most sheltered corners of the enclosure, putting some hay under the platform and perhaps some fake plants in front of it - there should be multiple exits but the birds should feel you can't see them when the are under the platform.
Shavings are a good substrate for the nesting area - it makes it easy for the birds to keep the eggs together. Some birds just won't go broody no matter what you do, some go broody all the time. If they are consistently laying eggs in the nesting area, you stand a good chance of them going broody. If they are laying the eggs all over the cage, the chance is not as good - but they might still collect the eggs in one place and go broody. If they are consistently laying their eggs somewhere besides the nesting area, they might go broody, but chances are they'll leave the nest later on because they don't feel safe.
Don't place food and water near the nesting area and don't place the nesting area near the door - your activities might scare the bird of the nest. Only check on the nest when the mother is not there - and do it as little as possible. If the hen has not gone broody when she has 12 eggs, remove all eggs and let her start over laying a new clutch - the eggs are less likely to hatch the older they are and the hen usually can't cover more than 12 eggs anyway. Sometimes they are very bad at covering the eggs the first time they are broody (I had one that couldn't even cover 5) and become better as they get more routine. They are also less likely to leave a nest if they have had chicks before. If you notice there are always eggs sticking out under the hen, remove a couple of eggs when she's not on the nest. If you don't, the eggs might go cold one after the other and stop developing.
Make sure you have a board or very fine mesh on the lower 5 inches or so of the enclosure - newly hatched button quail chicks can pass right through mesh that's less than ½ x ½ inch.
Also make sure you have low food and water dishes so the chicks can reach food and water, but place marbles, pebbles or similar in the water dish so the chicks don't get wet if they jump into the dish.
-To tell if your bird is broody, the most certain sign is she spends the night on the nest. A broody hen usually doesn't leave the nest for more than 30 minutes at a time and 10 minutes is more normal. She will leave the nest several times during the day, but spend the entire night on the nest. Also, since the hen spends so much time at the nest the droppings accumulate inside of it for a longer time and consequently they are much larger than usual - they can be as large as a quail egg. Sometimes the huge droppings is the first sign I notice when a hen is broody. Another interesting behavior I've noticed is that many hens will pick up a piece of bedding (shavings in this case) when they lay on the nest, turn their head and drop it on their back. The shavings then fall down behind or beside the hen or just stays on her back until she leaves the nest again. I'm not certain whether the hen is trying to camouflage herself or to build a nest, but I have only observed this behavior in broody hens. Another thing is the hen usually stops laying eggs when she goes broody. She might lay one or two more eggs after the first time she spends the night on the nest, but usually no more than that.
-Both parents raise the chicks, but be aware some birds have been incubator hatched for many generations and have no clue what to do when the chicks come. They might even attack them. So be ready to remove the chicks and raise them under a heat lamp if the parents aren't up to the task, or to separate the parent that causes trouble. Others have so strong parenting instincts they will raise any chick they see, even if they haven't been broody and the chick is not theirs.
Whew. I thought yours seemed to have most relevant info and that I wouldn't be able to add much more, but as I got started more and more things came to mind and now I'm not even sure I've got all of it