To me it looks like the hen was picked on by either other hens or a male or both, then probably got infected. Missing feathers on the back of the head are usually due to males grabbing the feathers when they mount. When my quail get into fights they usually go for the eyes; the injured quail usually ends up with a swollen eye but it's the actual eye area not below it like your picture shows.
I agree if you have some kind of antibiotic you think would work you should try it. Keep her separated from the others but if you can, try to keep them within viewing distance of each other. If they are separated too long the rest will consider her an outsider when you put her back and start attacking again.
I hope she gets better soon.
I agree if you have some kind of antibiotic you think would work you should try it. Keep her separated from the others but if you can, try to keep them within viewing distance of each other. If they are separated too long the rest will consider her an outsider when you put her back and start attacking again.
I hope she gets better soon.

