Absolutely! If a court has ordered you to pay support, it doesn't matter where you live, or where you move to - that court order is valid. Failure to pay could result in you having to pay the other side's attorney's fees, and even you spending time behind bars. Judges do NOT like it when their orders are not followed, and have a number of options to make you pay. And every time you don't pay is a separate judgment, meaning a separate time they can send you to jail.
File a motion to modify your support before you move.