@gamarus While I can't speak to 1.19 to 1.20, I did have a server that updated from 1.18 to 1.19 back in the day, and we were able to update it without any issues. You just put the mods in the required folder and delete out the old ones, and it should keep the information.
That said, definitely make a backup before messing with anything, and if it's giving you any trouble, you can always pull the world files down to your computer, update the mods, then reupload it back to the server (I had to do that a couple of times).
Good luck!
Edit: I just realized you were asking about Forge. Full disclosure, we used Paper, so your experience might differ. Sorry about that!