What is the similar meaning proverb in English?

Here is a related line from Shakespeare that has become idiomatic:

The lady doth protest too much, methinks.

It comes from Hamlet and you can read all about it here. As the Wikipedia article suggests, it has come to mean that one can "insist so passionately about something not being true that people suspect the opposite of what one is saying."

I have heard it in many forms, such as "methinks the lady doth protest too much," "the lady protesteth too much," or more simply "she/he protests too much."

The following proverb is a close enough fit to the situation described by the OP:

He who excuses himself, accuses himself.

The below proverb means that, a guilty person knows that he is guilty and no one needs to tell him that.

A guilty conscience needs no accuser.

Not a proverb, but I'm thinking of "The Tell-Tale Heart" by Edgar Allen Poe, a short story about a guilty conscience leading someone to confess.