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I'd give the sore leg a thorough check, see if you can find a reason for the soreness. And I'd separate them from the other quail and make sure they are warm and comfortable. If the one with the possible beak damage seems to have trouble eating normal feed, I'd give it something moist and soft. For both of them, if you have electrolytes, I'd put some in their water.. I'm assuming they are on a diet that has vitamins and protein in suitable amounts. Antibiotics might be recommendable, depending on what you find when checking the sore leg.
I don't have coturnix, so I don't know how likely they are to strain their legs by running around. But it's not something I've heard about before. If she is improving, that might be the cause though. If she stops improving or gets worse, I'd be worried. If the leg is warmer than the good leg in some places, that could indicate an infection, but as long as they have the same temp it shouldn't be the case.
Good to know the other one is doing fine :)