Al-Fagm was visiting a friend in the city's Shat neighborhood on Saturday when he got into an altercation with another friend Mamduh bin Meshal bin Ali.
State media did not give details about the nature of the dispute, but reported that bin Ali left the house and "returned with a gun and shot General Abdel-Aziz Al-Fagm," according to police.
A bodyguard was killed and two others, including the owner of the house, were injured.
When police arrive at the scene, Bin Ali "starts firing and refuses to surrender," police said. He was killed in the ensuing firefight.
On Sunday, the funeral of al-Fagm took place in the holy city of Mecca while the investigation into the incident continued.