I just started with my first hatch this year, hatching chicken eggs. I've never hatched a quail egg before, but I can tell you the basics for chicken egg hatching.
The eggs will take about 21 days to hatch.
The temperature should be 99.5 degrees Fahrenheit (forced air incubator) or 100 degrees Fahrenheit (still air incubator).
The humidity varies- if you ask 10 different people you will get 10 different answers. From day 18-hatch raise your humidity up higher then it was during days 1-18.
You have to turn the eggs at least 3 times a day. (I turn them at 8:00 in the morning, 12:00 in the afternoon, and 8:00 in the evening. But really it is up to you and what works best for your lifestyle).
You should candle your eggs on day 7, day 14 and day 18 to check air sac development. This will tell you if you have enough humidity, or too much humidity.
Here is a great diagram on what the air sac should look like: