This page here is to give you the low down on all you need to know about this particular item. Mostly why it is valuable and why I felt like pointing it out to you. The only items that get their own pages like this are ones that you should definitely be looting and ones that require "other" explanation other than just vendor it. For each of these items there are two methods of selling them, the quick and cheap way which is through /bar or if you mouse over an item in your inventory the hotkey for bringing up barter on an item is Alt + Shift + left click.
The other method is through the The Bazaar. To bring up the Bazaar it is Control + Shift + Left Click. Check the value of the item in both to determine which you'd like to sell it in. I have taken a picture of the barters up on my server, Fironia Vie, on 1/22/17 at about 9pm EST. I didn't take this for a price comparison, because your server will be different unless you're on FV no matter what. The reason for this picture is primarly to show you that this item definitely does have a demand. Also keep in mind that if you sell in The Bazaar you'll almost always net a MUCH larger price than to a barter.
What are Relic Fragments for?
Well, of course, when you find an item naturally you want to know what it is used for. Now that you know it can sell and it's valuable, time to tell you why! Relic Fragments can make a few different items such as Ring of Lost Seas, Thulisaur Mask, Arx Mentis Stud and Self-Regulating Rebreather and much more. It's the universal crafting material of TDS much like Ethernere Essence was for CoTF and Planar Energy Shards are for RoF.