Can I replay quests in Borderlands 2?

Solution 1:

Are you farming for quest rewards? Then, no. Once it completes, it completes for you. Only one reward per quest per play through.

You can run the quest again if someone else hosts the game and runs the quest (assuming they haven't completed it yet) but at the end of the quest, the game will tell you that you already have accepted the quest reward and won't give you another. (Your friend will still get the reward, but then they've completed it so they can't again, without another friend...)

If you're farming general item or boss drops that spit out on the ground for pickup, then you can by the above "friends' game" method.

Obviously, this is all for non-repeatable quests. For repeatable quests (they're marked in the quest log as such) they're, well, repeatable. So no issues there.