No, but having one stops the other one from dropping if the quest is not started and progressed to the point where you need to obtain the two pieces.
I did that mistake for many years, had one half on in my inventory and cleared MC weekly.
As soon as I took the quest and read on wowhead it dropped the next week, might only be circumstantial. However many people reported the same had happened to them, no getting both halves before having the quest.
Hmm… if this ever was a restriction, it certainly is lifted now. My mentioned Shaman got their first half five months ago, and the second half about a month ago. (I know, she was very lucky.) Without ever going out of her way to accept any quests related to the bindings, and still have not done so.
Unless, of course, the legendary quest line only is available for proper Sword wielders, thus the restriction still applies for those eligible for the quest. I am just speculating, and might actually go and see how eligible my Shaman actually is soon.