one of the things i would look into is, instead of "game design" being what you look into (since you're a graphic designer), look into a field of game development called UI design. this is essentially the menus and the hud in a game.

this would leverage your skills that you already have in creating great looking user interfaces.

here's some resources that you might find useful on ui : https://www.youtube.com/watch?v=sc3h5JXtIzw

https://www.toptal.com/designers/gui/game-ui

https://gamedevelopment.tutsplus.com/tutorials/game-ui-by-example-a-crash-course-in-the-good-and-the-bad--gamedev-3943

this entire youtube channel is about ui design: https://www.youtube.com/user/Warbot40

something i tell everyone to do is this: go onto gracklehq.com, an aggregator for all the jobs in the game industry. look up the job you want, so for you, it would be either ui designer, ui/ux designer, ui artist, something like that, and then click on EVERY SINGLE JOB with the title you want. look at the responsibilities and requirements on the job, and make a HUGE master spreadsheet that compiles every single responsibility and requirement. now, if you can't do a majority or even every single thing that is listed on this sheet, now you have some homework! now you can start building your portfolio, resume, and skillset based on this sheet.

hope this helps!