What happened was every time I went to where I was supposed to, the end of the quest would replay (the quest end scripting would initiate and some NPCs would kill each other) and I couldn't speak to the guy I was supposed to speak to because he thought that there was still a fight going on (there wasn't). So I was like, OK, I'll just talk to this dude they want me to talk to and it'll finish the quest.
I went to person B and they gave me the same quest that I had already completed, not realizing that I'd already done it. I think what was supposed to happen was that completing the quest for A meant that person 'B' was supposed to hate me and stop working with me (though the quest they were going to give me was identical). So, I received it from person 'A', went to where I was supposed to and completed it. This particular quest you can receive in two different ways.
Without giving away any spoilers, I was completing one of the last quests of the game.