Unless you have a CCW, you may not conceal a firearm upon your person.
As long as you are not concealing it upon your person, no permit is necessary. There is no distinction between whether you are in your home, in your vehicle, or walking down the street.
So carrying as you describe is perfectly legal. I had this same question and wondered if the seatbelt "concealed" the weapon. Or perhaps the seat itself? Or the center console? Or the door? Or... But it doesn't. Think about this. Am I carrying a concealed weapon if I'm standing on the other side of a wall and you can't see me? You can't see my weapon, but it's not concealed, it's just not observable from where you're presently standing. Remember, the legal definition requires it to be concealed UPON your person.
Hope this helps.