New Jersey is one of a few States that do not recognize or fully recognize the Federal Firearms Owners Protection Act (FOPA).
see here.
INTERSTATE DRIVING
Recently, a gentleman purchasing a car (flew in from out west) was kept in New Jersey overnight for some reason. He had not planned to stay in NJ, but something (lost bags I think), kept him from leaving the state. He had a firearm transported with him, in checked baggage. When he stayed overnight in the hotel, he was arrested for having an unlicensed firearm.
FOPA requires that you are legal to possess the firearm at your start point and end-point. Good luck with New Jersey. I think you are better off shipping the firearm to a FFL and having it officially transferred to the new owner.