Are you sure the gate opener is a Home Link system. I had this issue with my new Genie garage door opener. I had to use my garage door remote control to train the GX550 Home Link button first and then I was able to use the link button on the opener with that button on the mirror. You have to train the Home Link button on the mirror to not use Home Link basically and to uses a rolling code type system instead.
On the GX mirror the Home Link light shows orange when pushed if it is Home Link and green when it is not Home Link (rolling code). The instructions in the manual explain this but it is really not clear and I had to re-read it many times before I understood what it was saying to do. I had this same issue with my son-n-laws Toyota Tundra trying to get it to work with my garage door opener.
Read page 454 and 455 of the GX owner's manual carefully for the instructions.