Byron will start looking for you after you have received the Sword of Ancient Tongues at 550 displays and completed 'Shadows of One's Past'. Once he sees you in the world space, he'll run up to you and hand over a shard and then die. Three Morag Tong assassins are chasing after him and attack you. This will activate the quest in the quest log. In order to speed up the process simply travel to any area outside of Whiterun and wait for Byron. The sword itself does not have to be carried and can be left on display in the museum.