No, there isn't any evidence Hitler ever personally murdered someone Jewish. The only time it's strongly likely he killed anybody at all was in WWI. He was present as a regular infantryman during the infamous Massacre of the Innocents in 1914 (the First Battle of Ypres), when his own company was reduced from 250 men to 42. It's somewhat likely he ended up killing somebody there by law of probability, so absolutely horrific were the casualties on both sides. After that, when trench warfare started, Hitler had a job as a runner. He would be present at several major battles on the Wester Front, so it's possible he fired his gun, but he wasn't a part of a machine gun crew or artillery squad or anything like that which necessarily implied killing.

Hitler was also involved in some violent street confrontations in his early years in the party in Munich. While he eventually switched to trying to take power legally with a far more respectable persona after the Beer Hall Putsch, he started out his political career as a de facto terrorist in the shady world of postwar radical German nationalism. But the battles were usually against rival political parties.

As an extension, there's no evidence that Hitler ever had a personally negative experience with a Jew. Hitler himself does not cite one. His pre-1919 anti-Semitism, if it existed at all, was just your standard garden-variety prejudice you found all over Europe at the time. Mein Kampf had a political purpose in mind: Hitler strongly exaggerated his political views in Vienna.