Isn't your mom annoyed by the noise she makes? Tell her she does that because she's stressed out from being alone and getting another one would make her stop. It's not good for her to be alone - just like humans that are stressed are more likely to get sick, depressed and so on, the same goes for quail. Knowingly keeping an animal in a way that is causing it distress to a degree that is likely to shorten its life actually approaches abuse.
If your quail is female, a single female friend should be enough.
Is the bird cage free, since your mother thinks it's making a mess? Maybe you could try making/improving a cage that'll keep the mess better contained, to make sure your mother has nothing to complain about. And make it attractive so she can't complain about the looks either.