"Gendarmerie royale du Canada" is the French name of the Royal Canadian Mounted Police ("Mountie" if you prefer). The Québec Gendarmerie special forces would be the heirs of the RCMP Emergency Response Team. But heirs who were sold and turned into a subsidiary of Lone Star Security Services.
Canada main counter-terrorism unit used to be the RCMP
Special Emergency Response Team between 1986 and 1993, when it was disbanded and JTF-2 established (JTF-2 first operators were actually trained by SERT operators in SERT facilities).
I'm not sure JTF-2 would survive in the UCAS Order of Battle. Joint unit is not something American armed forces do (see Delta Force vs. Navy SEAL vs. USMC Force "SOCOM-won't-command-marines" Recon [1] ). So, in my opinion, they could disband the unit instead and turn the Canadian Special Operations Regiment or a recreated Canadian Airborne Regiment into a Tier 1 unit. Or recreate WW2 1st Special Service Force, which would actually be quite a smart move from a symbolic point of view.
[1] For the record, since 2006, the USMC officially has special forces under SOCOM, the
Marine Special Operations Regiment.