Yes you can eat quail egg shells and they are healthy for you.
When eating quail egg shells you should grind the quail egg shells up into a powder and then eat them that way otherwise it can be hard to chew the quail egg shells and they could pose a choking hazard when eaten without grinding them up.
The quail egg shells are a great source of calcium as well as other nutrients and minerals that our bodies need to stay healthy.
The quail eggs themselves are also very good to eat and health although quail eggs are more expensive than chicken eggs because there's not as many mass produced quail eggs as there are chicken eggs.
Quail egg shells contain 90% calcium carbonate and contains lots of essential minerals for the body, like copper, zinc and lots more.
Chicken egg shells can also be eaten as well and you'll get calcium and other nutrients and minerals from those as well.