I am wondering what expressions are used for this because I need to describe a person for a school essay and this is their personality.
Fake self-confidence is often described as false bravado. A person can be full of false bravado in a particular moment, or they can be that way habitually.
If you want an actual idiom, there's already a related question about people who boast too much with a good list in the answers. That's more about people who are all talk and no substance, which is not necessarily the same as projecting false confidence, but one of those expressions might still work for your description.