Maybe. In January 1787 young Beethoven travelled to Vienna and we can safely assume that he wanted to become Mozart's pupil. He stayed there until the end of March. Then he travelled to Munich, visited Regensburg and Augsburg and by the end of April he quickly returned to Bonn, most probably due to a letter of his father about the very bad health of his mother (she died in July 1787). Beethoven himself never said a word about a meeting with Mozart, nor did Mozart, nor one of those who may have been present. According to Czerny Beethoven had told him that he had heard Mozart playing the piano (and he didn't like it, too "staccato" and "old-fashioned"). However, according to another testimony Beethoven had met Mozart, but never heard him play. Yet we all know that famous story about Mozart praising young Beethoven loudly. Beethoven's friend Holz, who met him for the first time in 1825, was the one who threw the story into the world after Beethoven's death and we don't have any knowledge about the source.
See:
Raab, Armin. Beethovens Mutter - Legenden und Tatsachen. In: Bonner Beethoven-Studien. (Bonn, 1999).
Habler, Dieter. Beethovens erste Reise nach Wien. Die Datierung seiner Schulerreise zu W.A.Mozart. In: Neues Musikwissenschaftliches Jahrbuch (2006).