Push Notifications don’t really exist for Lemmy yet, as they aren’t supported in the backend currently:
https://github.com/aeharding/voyager/issues/1027 https://github.com/LemmyNet/lemmy/issues/2631
As someone else here said, your best bet is probably monitoring your Inbox RSS feed.
Magic Earth (while not FOSS, it’s privacy oriented though) can do this.
There’s also Transportr, but AFAIK it’s been unmaintained for quite a while now, so it may not work / stop working soon.