I was thinking it meant "I believe I can be happy"
put on hold as off-topic by naruto, Dono, ajsmart, broccoli forest, Blavius Sep 13 at 17:40
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If I just read it literaly, it does mean what you said at least. "幸せになれる" means "I can be happy" just like "と信じています" does mean that "You are believing" so it will means "I believe I can be happy in the future". I would like to put "in the future" because it does sound like th person really want to be happy on a larger side than just "I can be happy".