I think you can but you still have to link to Google play services, which is not possible for open source apps in f-droid since they need to be able to run on de-googled android which lacks Google services.

It looks like FCM is supported by the open replacement I linked in a close-by comment, here: https://news.ycombinator.com/item?id=23571663

Aren't microg supposed to get installed first on the device? Which mean if the user doesn't have it, then your app won't run? Or can you embed the FCM support into the app itself so it doesn't depend on google play service or microg availability?

You're asking the wrong person. I'd have to research just as much as you'd, and judging by upvotes and age, (almost) no one else would care. Sorry :(

I found no evidence for either side, but I can't easily test it right now.