I'm going to throw my two cents in here for what it's worth. Only because I've hatched a lot of button quail this year and through trial and error have found what works best for me.
First of all, I'm using one of those cheap styrofoam still air incubators. Not the best incubator but it's been doing a fine job now that I have it down.
The thermometer in your incubator needs to be at about the same level as the top of your eggs. I've seen some people say to wash the eggs before you incubate them. I don't recommend this at all as you're washing the protective barrier off the eggs allowing bacteria to possibly infect the eggs during incubation.
Get some of that soft rubber type shelf liner, the type that kind of woven and can breathe and line the bottom of the incubator completely with it. You don't want your buttons little legs to fall through the wire.
Very important, place your incubator in an area where the temperature of the room stays fairly constant. Room temperature fluctuation will have an effect the temperature in the incubator. You want to keep the temp as constant as possible. I have a large walk in closet off my master bath and that's where I keep my incubators.
Set your incubator up 24 hours in advance to give you plenty of time to get the temperature right.
I don't have anything to measure the humidity in my bators, but what I do is I keep the water reservoirs in the bottom filled at all times and I also set a coffee cup full of water in the bator. This seems to work perfectly.
As for temperature. I now set my temperature at 99.5. It's been my experience that hatching at temperatures above 100 results in weaker chicks and more foot and leg problems. If you set at 99.5 you pretty much eliminate the possiblility of the temp spiking too hot. Falling a little below 99.5 for short times doesn't hurt them. But spikes over 100 aren't good. Trust me on this one. When I started I was told to set the temp at about 101 degrees. This didn't work for me at all. Many of the chicks were weak and had leg and feet issues and died though at the time I didn't know why.
The first couple of batches of Button quails I hatched I used an automatic egg turner. My hatch rate was only a terrible 10 to 20 percent. For the following hatches, I removed the turner and turned the eggs by hand. You can mark the eggs if you want to, but it's not needed. Lay the eggs on their sides in the middle of the incubator and when you turn them just lay your hand over them and gently roll them. Do this at least twice a day. Three times is better.
After hatching, leave the chicks in the incubator as long as possible! If you take them out too early you will have problems. I now leave mine in the incubator for as long as 24 hours and the result is stronger chicks, ready to eat and drink, and a lower mortality rate.
For a brooder, I use either a 10 gallon aquarium or a plastic rubbermaid container, lined with the same shelf liner so they don't slip and hurt their legs, with a brooder lamp over them. This is the trickiest part for me. Set the light over only half the brooder and make sure it never gets over 95 degrees under the light. You will have to adjust the light up and down many times to get this right so start BEFORE you place the chicks in it until you get it right.
For the first few days I use only a very shallow lid from a water bottle or jar or whatever you can find. Make sure to check it often to make sure they always have fresh water as it can evaporate quickly. I feed them game bird starter that I throw in my blender and blend to a fine consistency and at first I feed them directly on the floor, again, because it is so easy for them to throw their little legs out, I'm afraid they would slip in a dish and hurt themselves.
Since I have been hatching and brooding my buttons this way, I have not had a single button with leg or foot problems. Whereas before, I would have at least two in every hatch. Sometimes more.
As others have said, tap your finger lightly in the food and near the water to teach them to eat and drink.
Sometimes you will get a baby or two who don't quite get the eating drinking part down right away. Water is more important than food at this point. If the baby won't drink on it's own, gently dip it's little beak in the water a few times to make sure it gets some water. Do this a few times a day until it starts drinking on it's own. And keep trying to coax it into eating by gently pecking the food with your finger to mimic a hen pecking the ground. As long as you keep the baby hydrated, he should begin eating in a day or two. It doesn't take long for a chick to die of dehydration.
I've had chicks who were close to death, to the point that they would fall over on their backs and not be able to get up and walk and managed to save them by forcing them to drink. So don't give up on any of them.
Good luck. I got up this morning to a bunch of new button chicks in the incubator. And they hatched right on time.