The G stands for Grisha.
When Callen was in Russia to break Arkady out of prison on "Matryoshka, Part 2" (Season 7, Episode 16), he met a local man named Garrison—who we recognize as the man from the cafe in Season 6. During their encounter, Garrison says that Callen's mother wanted him to know where he comes from. We knew that the name "Callen" is from his mother's side of the family, but the G has long been a mystery. Garrison revealed that it's short for Grisha, which comes from his longer name: Grisha Aleksandrovich Nikolaev.