Why can't I junk certain jewelry?

I know while playing, the game has not let me sell, destroy, or put in my junk bag any quest items, so I'm guessing that's the cause. While I don't have firsthand knowledge of your particular item, I did find some info on it.

Brother Fallon’s Beads

St. Hadwyn’s Beads is an accessory which you’ll get in the Plaguseshield. You will not be able to thrash it, go to St. Hadwyn Mission (Glendara) and talk to Brother Udo. Make sure that you make him certain that you believe in the ways of Mitharu in order to trigger this quest.

So it sounds to me like it's a quest item for a quest you don't have yet.