What happens to the Qui-Gon Jinn lightsaber is that it eventually gets passed down to Obi-Wan Kenobi when Qui-Gon Jinn dies. Obi-Wan would go on to use this lightsaber until his death at the hands of Darth Vader. The lightsaber then goes into the archives with all of the other lightsabers with its uniquely distinctive design. Qui-Gon Jinn was the only Jedi that used this design.