He has started training with his new teammates at Scotstoun, after arriving in Glasgow from Cape Town.
Thomson has played six times in Super Rugby for the Stormers and has scored 26 points.
He has also represented South Africa U20s at the World Rugby Under 20 Championship and has played for Western Province in the Provincial Cup and Currie Cup.
Speaking to glasgowwarriors.org, Brandon said: "I was over in Zimbabwe playing a warm-up game for the Stormers when I got the call to say I could fly over here and it all happened pretty quickly.
"I came over with the main goal of getting some experience and it's good to play in a different environment and for a great club.
"I'm structured, physical and I like to run the ball out wide and keep the game loose.
"My father is from Scotland, so it's nice to be able to come over here to play."
Glasgow Warriors head coach Gregor Townsend said: "It's a bonus for us to be able to bring in someone of Brandon's quality and potential during a period when we have a number of players away on Test duty.
"He has fitted in really well so far and even training in the snow yesterday didn't seem to phase him!
"We look forward to seeing how he performs over the next few weeks and we'd like to thank Western Province for their support."