The sidequest Lost Relic is only available after you have recruited Florentius Baenius into the Dawnguard during the quest "Bolstering the Ranks".

Talk to Florentius inside Fort Dawnguard. Ask him what you can do to help. He tells you he has been pouring over some of the tomes left here in the keep. Interesting stories, indeed. Seems the previous incarnation of the Dawnguard had, at one time, some interesting toys. Arkay and him think it could be a great advantage for them. You should look into retrieving it.

This starts the quest:

Lost Relic
Florentius Baenius has sent me to retrieve the Dawnguard Rune Hammer from Greywater Gorge.

Retrieve the Dawnguard Rune Hammer from Greywater Gorge

If you ask him what the artifact is, he tells you it is a powerful warhammer imbued with magic that places wording runes on the ground to obliterate it's wielder's foes.

Note: The item you are asked to retrieve, and the destination, will be random. It could be a Dawnguard Rune Shield or Dawnguard Rune Axe instead of the Dawnguard Hammer. Regardless, the quest will be the same. You just have to go to the assigned dungeon, fight your way through the end, and locate the artifact, most likely inside a chest.

Once you have retrieved the artifact, return to Florentius in Fort Dawnguard. Tell him you have retrieved the Dawnguard Rune Hammer from Greywater Gorge. You get to keep the artifact.

This completes the quest, "Lost Relic". Talk to Florentius again to see if he has additional quests for you.