As far as I know, quail don't pluck their feathers when going broody. I only have buttons but they certainly don't and none of the stories I've read about broody coturnix mentioned feather plucking, so I'd search for a different reason.
I guess we're looking at 4 options:
1) She plucked the feathers. Only an option if she can reach the entire area that has feathers missing. She could have done so due to feeling irritated by parasites or something internal - like a tumor or an egg that's stuck, for instance. A tumor or egg could also explain har walking strangely - it could be applying pressure to the nerves to her legs.
2) The others plucked her feathers. Can't see why they'd pluck her chest though, unless her chest feathers looked very strange.
3) A predator caught her feathers. She could have hurt her leg when escaping. Only possible if the shed isn't fully predator proofed.
4) She flew/walked into something that caught/scraped of her feathers. As she doesn't fly with the chest first, it would probably have happened during landing if it had something to do with flying. And she could have hurt her leg at the same time. Tro to look for places in the enclosure with lots of feathers.