You had some showdown value with your two pairs, so I think a check behind would be the best approach for the river. Since you decided to bet, your bet was way too small relative to the pot size, so you were not really making him fold a better hand. Also, your bet was too small for someone who had a Q, you would try to get more value for such a good hand. The way the hand was played makes it difficult to realize if the other player has you beat or if he figured out you didn't have the nuts and is trying to bluff you. His actions could also mean he was going for the flush and is now trying to bluff you. If you had checked the turn he could be more inclined to bet the river and you could better evaluate your options.
This is a very difficult decision and, in your position, I would always check behind this river. In any case, I would probably fold this hand and try to get myself back up in better spots.