Initialism of maternal grandmother.
"In her family tree diagram, she labeled her mother's sister as mgm to indicate their relationship as maternal grandmothers."
In plain English: Mgm is an old-fashioned way of writing MGM, which stands for Metro-Goldwyn-Mayer, a famous movie studio known for making big Hollywood films.
"The MGM logo flashed on the screen before the movie began."
Initialism of Metro-Goldwyn-Mayer, a US movie studio.
"The MGM logo featuring the roaring lion is instantly recognizable to film enthusiasts worldwide."