Yes, but a lot later; dunno when, no idea, but he stayed in middle-earth for some more time after the war of the ring and re-established the realm of Mirkwood, renaming the region Eryn Lasgalen, Forest of Green Leaves... he stayed together with Celeborn who ruled over parts of Mirkwood called eastern lorien
Im quite sure it was Legolas, but I dont know when Thranduil left;
And I also dont know if they left together, but seems logical;
After Galadriels parting he grew weary of middleearth and moved to Rivendell, as he is said to have left middleearth last of all elven lords, he probably went after Thranduil, though I dunno if he counts as elven lord.