It's best to wait until they're a few weeks old (about 4 weeks). You can start them in a smaller solid floor brooder as long as you watch to make sure they don't get too crowded (they grow really fast).
i put mine on paper towels for the first week. then move them onto aspen wood shavings(any wood shaving/paper anything will work, just not cedar wood). then at week 4/5/6 i'll put them in the grow out pen that has a wire bottom.
I usually put coturnix on wire at about a two weeks of age... I run them inside, in a tote brooder for two weeks. while inside, they are on wood horse pellets covered with paper towel, or just a old cloth towel. Its a shock to them when moved to the wire pens in the shop.(with a thermo controlled heat lamp) But they adjust fast... I'v done 100's this way with no problems...Bobs and chukars too...
As Bill said, two weeks is fine to start Coturnix on wire. If your 2 week old birds lack the strength or size to navigate 1/2" hardware cloth, it's time to seriously rethink the line of birds that you are raisin'. There are plenty of quality, larger size, strong birds out there. You just have to look for them, and don't settle for less. I wouldn't own any Coturnix quail that need 4 to 5 weeks to be able to safely navigate 1/2" hardware cloth.
Yeah I have two different sources of quail chicks and introduced them all to wire at 2 weeks or so, if you let them stay on bedding they will make a mess of it in no time.
The way I have mine set up is a 4ft x 4ft cage that is has a drawer under it, I put bedding in that to keep the smell down I put pine bedding in there, every few days I take the tray out and change the bedding so their feet never get covered in feces and they get used to wire which will be their permanent home eventually.
When I kept mine in a tote brooder in bedding they were CONSTANTLY knocking their water over and getting their bedding wet and musty, it cant be good for their lungs or ours.