The photo attached is the jumper-wires-everywhere-proof-of-concept prototype. The real thing looked pretty much the same but with less breadboard and more duct tape.
I don't have any photos that aren't super close up (faux artistic >.<), unfortunately. I kind of felt like no one wanted to see my ghettotastic quail mother substitute. I do have lots of cute egg and fluffy chick photos, though, if you're interested!
I can explain to you how I did it, though:
First, I they hatched in July last year which was a hot and humid month. This was great because it helped guard from drafts and temperature fluctuations. My container was a Rubbermaid bin wrapped in a blanket. If I did this in the winter, I'd probably build it out of wood or thick walled styrofoam. Likewise, I wouldn't leave it open since it gets really dry whenever it freezes. The lid was open and had a hardware cloth screen over it, and suspended over the screen was a heat lamp. One of those bowl reflector shaped deals with a clamp (I think they're $11 at Home Depot? This was just a leftover from a previous half assed attempt at growing weed), and this bulb:
I didn't end up using the bulb in the photograph because it was white light and I figured it would keep me awake.
I got a Grove relay from RP Electronics for the lamp:
http://www.rpelectronics.com/se-act05161p-grove-relay-module-spdt-30a-125vac.html ($7 after my discount)
which was pretty slick because the operating voltage is 3.3V - 5V, and it's got a cute little pinout and the terminal block.
If you are using a thermostat, you'll probably need a relay, unless there is one inside of it already. Either way, remember that the relay goes on the LIVE wire of the lamp, NOT the neutral wire.
For temperature and humidity sensing, I used the DHT11, http://www.rpelectronics.com/dht11-dht11-digital-temperature-and-humidity-sensor.html ($5 on eBay), soldered long wires on, and found a nice central point to stick it. It's got to go near the eggs basket(incubator)/bed basket(brooder). If you turn by hand, just mark both sides and keep them all together in a shallow padded dish. Then you can just stick the sensor directly underneath the centre spot in the dish. Also if you keep them in a dish they'll be a bit more earthquake proof. ;)
I used an Arduino nano for the final project since at $1.50 each they're pretty disposable, but I borrowed the analogue button/LCD shield that I use for prototyping because I was too lazy to spend half an hour soldering buttons to a protoboard. (They're about $15 at RP Electronics.)
The computer fan seemed to be happy at 6-12V which was great because I ran the whole thing off a 9V power supply from ye ol' drawer of power supplies scrapped from dead electronics. The fan just piggy backed off of the Arduino power supply. I just placed the fan over the screen peering down into the box at a 45 degree angle and secured this with a flap of cardboard from a case of beer and a lot of duct tape.
A bowl of water was placed on the bottom as close to the lamp as possible. The humidity never bloody dropped below 70% though, in the brooder or my house. It was so gross and sticky.
Programming the Arduino was pretty simple, and it only took me one growler of beer to finish it. The logic was basically that the lamp should turn on if the temp dropped half a degree below the set temperature, and off if it was half a degree above. This tiny bit of leeway kept the lamp from flickering on and off constantly since the DHT11 is sensitive to 0.1 degrees. I also programmed the temperature set to adjust in increments of one degree which is important for the broodery aspect of this.
After they hatched, they slept in there for about two weeks until they finally decided they rather nap on my chest or in front of the vent on my Minecraft server. Two years later, Pumpkin is STILL stretched across my chest. I guess it was a pretty mediocre brooder in the end because it didn't perform the most essential quail rearing activities like kissing little feathers and whispering "you are sooooo cute, I love you so much. <3" in their ears.
If I did it again, I'd definitely do it in a hot/humid month for the extra security, and I'd also probably use a proper reptile terrarium heat lamp because it covers so much more real estate than the spot lamp. I also might use the automatic egg turner my sister built out of K'Nex if I'm taking summer classes at the time, though I'd probably disconnect it sometimes because I still think it's good to bond with your eggs and talk to them whenever you pick them up so that they recognize the sound of your voice.
I probably could have made it look a lot slicker and used less crap from around the house to build it, but as cheap and dirty as it was, it was successful!
Hopefully you can take a bit of info out of this that will be useful to you. Since you seem to have most things, you could probably get away with purchasing no more than an Arduino and a relay then hard coding the temperature in. If nothing was helpful, don't worry! I might do an incubator/brooder workshop at VHS in late Spring and/or make an Instructable.