We cannot leave out that “that” because it is the subject. We can leave out “that” when it is the object, as in “There is a proverb (that) I know.”

no，if without "that", you have to use "saying" instead of "says"

There is a proverb. The proverb says to be a nice person. One sentence: There is a proverb that ( = the proverb) says to be a nice person.

No