I only practice federal law in Florida, but I think it will show up. The juvenile's record normally would be available to all law enforcement unless it was removed at a later time.
It may not show up, why not give it a try and find out. You can also just disclose the conviction along the lines of "I was charged with a drug offense when I was 13." I doubt anyone is going to get to exercised about that.
You may go to 1 Police Plaza nd ask for certificate of good standing, pay a small fee, run your finger prints and get an accurate print out of your criminal record, as it would show through FBI records.