Living in the Garden State I can tell you that you will be okay, provided the firearm is unloaded and separate from the passenger compartment.
There are so many false stories going about regarding New Jersey and hollowpoint ammunition. I think these stories can be traced to an article written by Massad Ayoob 10 or so years ago.
One of his students from Connecticut got caught visiting NJ carrying hollowpoints in a S&W Model 60. The person did not have a NJ CCW license. Since this person was from out of state the judge called for $5,000 bail. The Smith carries 5 rounds.......$5,000 divided by 5 = $1,000. A NJ resident would have been given a $500 bail (10% of $5,000).
Obey the traffic laws and you will be okay.
