Unlocking the quest
The quest can be unlocked in two basic ways. The first one is to defeat Thane during [The Maker Warrior] side quest. After winning the duel, you can receive a commission to kill Bheithir along with three other monsters. The second way is to find Bheithir without the involvement of Thane, at which the quest will automatically appear in your quest log. You won't have to take the quest from Thane first, but simply kill Bheithir and go to the Ancient to receive your reward.
Find and kill Bheithir
Bheithir is pretty easy to find. Nevertheless, given that he's a fairly difficult enemy that can deal a lot of serious damage, I wouldn't recommend going into battle too quickly. Bheithir resides in [The Nook], a small dungeon connecting Baneswood with the Lost Temple. You need to enter an underground cave and use interactive handholds to get across to a small bridge. More details are described in the chapter detailing the Nook.
Return to Thane
Return to [Tri-Stone], initiate a conversation with Thane and tell him you killed Bheithir. The primary rewards for this side quest are 603 XP and 3000 gold, although killing Bheithir also gives you a new rare weapon, Bheithir's Talons.