In Civ 4, as I recall, you could trade for another civilization's vote in the U.N., but I can't seem to find that option in Civ 5. Are the only votes I can earn those of the city states?
The manual explicitly states that all AI civs will vote for themselves unless they have been liberated. So the only way to get them to vote for you is to have their capital be captured by another civ, then have you conquer it and give it back to them.
I believe if you're on friendly enough allied terms with an AI, you could get their vote. There does not seem to be a set-in-stone "I give you my vote" diplomatic agreement, however.