From Firearms Frequently Asked Questions (FAQ's) | New Jersey State Police
25: Can I travel through New Jersey with my firearms?
Yes. With no unreasonable deviations from your travels you may drive
through New Jersey with any type firearm or ammunition. If New Jersey
is the State of destination, then the firearms must be legal in this
State.
and
6: How do you transport firearms?
Firearms shall be carried unloaded and contained in a closed and
fastened case, gunbox, securely tied package, or locked in the trunk
of the automobile in which it is being transported. If the vehicle
does not have a compartment separate from the passenger compartment,
the firearm and or ammunition shall be in a locked container other
than the vehicle's glove compartment or center console and kept
farthest from the driver.
As a precaution, ammunition shall be transported in a separate
container and locked in the trunk of the automobile in which it is
being transported. Ammunition magazines may remain loaded during
transportation and shall be carried separate from the firearm.
Unless, of course, the gun is a type simply illegal to own in NJ, such as an assault rifle, automatic rifle, etc. This may be dated, but outlines what firearms are illegal: A Guide to Legal Firearms Possession in New Jersey - HG.org.
See also New Jersey State Police - NJ Firearm Laws - N.J.A.C. Title 13 Chapter 54 - Firearms and Weapons - njac-title13-ch54.pdf