Talk with either of the brothers about getting some work. They will mention that they do not believe in the stories of the dragons that have been sighted around Skyrim. They want to see the Dragon (and fight it) for themselves. He will mention that they have heard of one of the Dragon Lairs in a Hold you have visited. Head there.
When you arrive, you will need to track down the dragon. If you head to the shrine, they will occasionally seem like a statue. If you see something that looks like a dragon, but is not moving, shoot it with an arrow. That will trigger the fight and get the dragon after you. Defeat the dragon with the help of either brother.
Talk with them after the fight to complete the quest.