It’s a waste of time on their part. Selection of the GG is a federal power exclusively, and made by the monarch based on the advise of the PM.
Language doesn’t enter into it, beyond the convention of alternating between anglophone and French GGs.

Edit: I remember the hoopla when her appointment was made and a number of complaints made to the Commissioner of Official Languages, who investigated the complaints at the time.

Here’s the preliminary response:
https://www.clo-ocol.gc.ca/en/newsroom/2021-11-26/statement-commissioner-official-languages-canada-preliminary-investigation?wbdisable=true

One of the important bits:

Furthermore, the Prime Minister, who is not a federal institution subject to the Act, has full authority to appoint any person to the position of Governor General and is not bound by any recommendations issued by the Privy Council Office. For all of these reasons, the conclusions in my preliminary investigation report on the process that led to the 2021 appointment of the Governor General are that the complaints are unfounded.