As your possession of the gun once you arrive in NJ will be prohibited, you're not covered by the Federal law.
NEW JERSEY--New Jersey laws governing firearms permits, purchaser identification cars, registration and licenses do not apply to a person who is transporting the firearm through this State if that person is transporting the firearm in a manner permitted by federal law. This federal law permitting interstate transportation of a firearm applies only if all of the following requirements are met: possession of the firearm was lawful in the state in which the journey began; the possession of the firearm will be lawful in the state in which the journey will end; the person is transporting the firearm for a lawful purpose; the firearm is unloaded; the firearm is not directly accessible from the passenger compartment of the vehicle; the ammunition is not directly accessible from the passenger compartment of the vehicle; if the vehicle does not have a compartment separate from he passenger compartment, the firearm and ammunition must be in a locked container other that the vehicle`s glove compartment or console. A person transporting a firearm through the State of New Jersey in the manner permitted by federal law need not give notice. Whenever an officer has probable cause to believe that a person`s possession of a firearm is in violation of New Jersey law and not permitted by federal law, then the law enforcement officer may make an arrest.