The Juilliard Youth Symphony was a well known group of musicians.
Prior to serving on USS Voyager, Harry Kim played the clarinet in the Juilliard Youth Symphony, a fact Captain Kathryn Janeway relayed to Tuvok after Kim had gone missing. She was worried that she never had the time to get to know her crew anymore. (VOY: "Caretaker")
While there is no symphony with that exact name today, the term "Julliard" brings to mind the prestigious Juilliard School of music in New York City.