Anything that's unconvincing is unlikely or hard to believe. Your unconvincing story about being abducted by aliens probably won't persuade your teacher to give you an extension on your term paper.
Your unconvincing impersonation of President Ronald Reagan might sound more like John Wayne, and an unconvincing alibi won't get a suspect off the hook if a detective thinks she's guilty. The adjective unconvincing comes from convince, "cause to believe in the truth of something," and its root that means "to conquer." If you haven't convinced someone of some fact or truth, you've been unconvincing.