Can you refrigerate unbaked quiche for baking the next morning?
The crust would get soggy.

If you want to make a quiche in advance, you can make the shell and blind bake it, if required.  Wrap the shell in plastic wrap and refrigerate it over night.

You can make the filling custard and have it in a bowl (I would actually use a sealable container).  Eggs and cream are highly perishable, so you want to refrigerate them right after mixing.

Then, to bake your quiche, all you need to do is pour the filling into the shell and bake.  

The Kitchn and Fifteen Spatulas also say you can pre-bake your quiche and then reheat it the next day for service.  Fifteen Spatulas even claims the flavors meld and improve.