When talking to one of the two lords, you can agree with him, as to take his side in the argument. This will resolve the dispute, but the other lord will think you have taken the other's side. If relation is high enough, however, you can ask the lord to "settle his differences". You must have enough relation with the other lord to do the same with him, which will conclude the quest with no conflict.
When given by a spouse, they will request furs to make a velvet robe to please your rival, or someone who dislikes you. After that, you may speak with the lord and complete the quest.