I can tell you that mere possession of any handgun in New York State without a New York permit is illegal. Even in your home, let alone a vehicle. Even if it's locked up. You WILL get prison time in NY if you're caught with it.
Plus, there can be other tricky problems. In NJ, possession of hollow point bullets by anyone but a cop is illegal.
If you want a gun for self-defense, you are far better off with a pump shotgun, or a lever action cowboy carbine. General rule of thumb is that the more a gun looks like Grandpa's hunting gun, the more socially acceptable it is, and the less likely it is to be legally problematical. You could even take either of those into Canada, just fill out a simple form at the border.
Otherwise, if you want a handgun, you've got to be SURE of the laws in each state you pass through. And you may have to detour around certain problem states.