"Prove me" / "prove to me" / "confirm one's belief"

The second sentence, 'prove to me', is more natural (and indeed correct where the first one is not). Prove is a verb so needs the to.

The 'confirm my belief' (me in the belief isn't quite right) sentence is a bit formal.

The verb prove is never used with an indirect object, only a prepositional object.

You can use indirect object with verbs of communication and and giving, such as

He asked me a question. She gave me a dirty look.

But you would never say

He proved me an equation. She proved the company her value.

The correct form for that is always "proved (direct object) to (prepositional object)", as in

He proved an equation to me. She proved her value to the company.

The only case where you would find a personal noun or pronoun directly after "proved" is when you are using that personal reference as a shortcut to mean something that this person has said (and therefore they are the direct object of proved):

Prove me wrong, if you can.
The course of events proved him correct.