16-11-128.
(a) A person commits the offense of carrying a pistol without a license when he has or carries on or about his person, outside of his home, motor vehicle, or place of business, any pistol or revolver without having on his person a valid license issued by the judge of the probate court of the county in which he resides, provided that no permit shall be required for persons with a valid hunting or fishing license on their person or for persons not required by law to have hunting licenses who are engaged in legal hunting, fishing, or sport shooting when the persons have the permission of the owner of the land on which the activities are being conducted; provided, further, that the pistol or revolver, whenever loaded, shall be carried only in an open and fully exposed manner.
This code section does not apply to your motor vehicle. I've highlighted it with blue.
What you are interested in is in 16-11-126. Specifically section d)
16-11-126
d) This Code section shall not forbid the transportation of any firearm by a person who is not among those enumerated as ineligible for a license under Code Section 16-11-129, provided the firearm is enclosed in a case, unloaded, and separated from its ammunition. This Code section shall not forbid any person who is not among those enumerated as ineligible for a license under Code Section 16-11-129 from transporting a loaded firearm in any private passenger motor vehicle in an open manner and fully exposed to view or in the glove compartment, console, or similar compartment of the vehicle; provided, however, that any person in possession of a valid permit issued pursuant to Code Section 16-11-129 may carry a handgun in any location in a motor vehicle.
So, you can either have it open on the seat or you can have it in your glovebox or console.