Police said the driver got in an argument with a man standing outside the vehicle shortly before 6pm on Sunday.
A spokesman said it was believed the man outside the car was shot before the man in the car drove off.
The injured man was taken from the carpark, near a group of fast food restaurants on Nancy Bird Way, away from the terminals, to the Royal Brisbane and Women’s Hospital.
Police said he had non-life-threatening injuries and there was no ongoing threat to the public.