Kill the Beast for the abbot in the crypt of Thule. It's lair is in the south of the crypt, at the end of a secret tunnel. Only one weapon can kill him, the Sword of God, The abbot hid it on a shelf near the entrance to the lair and walled up the shelf.
You will need a rope to complete this quest.
Go to the Thule Cemetery.
Go down into the Crypt
Talk to the abbot and say you'll kill the beast for him.
He'll give you the crypt key.
Go to the Southeast part of the Crypt.
Use your rope to advance through the tunnel.
When you reach the door at the end of the tunnel, hit the wall to your left and get the sword of god.
Equip the Sword of God and use the crypt key to open the door.
Kill Boad the Beast.
After killing Boad the Beast, take 1 step south, 2 steps East, and 1 step North, and touch the fountain.
It will say "You drink from the strange looking, ice cold waters. It clears your mind and fills you with tranquility. You've gained back your lost experience! 1800 experience has been restored. (The amount restored may differ)
Then go south towards the hammer on the wall. It will say "Hmm. Perhaps this hammer has a special use."
Take the Redshift Hammer.
Return to the Abbot.
He will tell you the whole truth and you'll find out the governor was a part of it.
He says to go take care of the Governer quickly because he's planning to leave Eldfell soon.
He also gives you the abbey keys and says to give the keys and the Sword of God to Brother Joram who will replace him.
The Kill the Beast quest is solved and your Fame has increased.
If you want the good ending, select option 1 that says "I'll take care of the Governor.".
If you want the bad ending, select option 2 that says "If I kill you, no one will know the Beast is dead. The Governor and I could be partners!".