It sounds like she could be egg bound, but I've never actually seen this myself. She could have picked the feathers herself due to distress and pain in the area. I believe people place the birds in shallow, warm water to make it easier for them to pass the egg. And if that's not enough, they might put some kind of grease on and in the vent.. But if she is egg bound, you should be able to feel the egg in front of the vent. If you can't, she might be the one that laid the egg with blood on it and she might have had trouble passing that one, causing her current appearance. You might want to apply some antibiotics as she could have internal damage.
Now, to making sure it doesn't happen again - are you feeding them additional calcium? Egg binding and eggs without a shell can both be consequences of a diet that has too little calcium. Oyster shell, grinded to a size the quail can handle is a good way to supply calcium.
If they do have calcium available - do they get direct sunlight or does their feed contain vitamin D? That's important for them to be able to utilize the calcium.
If that's not it either - what are you feeding them?