Can a belief be a fact?

Well, they are misspeaking. Belief cannot be fact. That, however, is not to say that belief cannot be true. Truth, in life, can be composed of facts, but if we put the facts in different orders of priority because you and I are different people, our truths are going to be slightly different. To make a metaphor of science, I'd say that hypothesis are beliefs, facts are data, and truths are theories. I never forget the "fact" that Myth has broader shoulders than mere facts, for it often bears a greater burden of truth.