Overread
Searching for a flower
No, unicorns don't crop up anywhere in The Silmarillion or LotR. I suspect (though I'm not certain) that they represented a type of allegorical symbolism to which JRRT took a rather dim view....
interesting - I guess we are so used to so much of fantasy being in LORT that its very easy to think some common fantasy creatures are in there when they are not!