Yeah. A quail could injure themselves on the beams across the top. I would also recommend using half inch hardware cloth. It is better for keeping predators out. Everything likes to eat quail.
That looks like she flushed and broke her neck. She may have done so before but not as hard, which would explain why she seemed injured. How high is the roof of your pen? It should be either too low for them to get up enough speed to harm themselves when they flush or high enough that they...