So the protocol is way, way different and massively out of scope to actually reimplement, so it would never make sense to have chugging along within the Lemmy backend server itself.
However, embedding Matrix rooms in webpages is something the Matrix devs want to make more straightforward (Gitter does this nicely and they're shooting to subsume all of its functionality) so it's not too hard to imagine some kind of integration with a. a separate Matrix server that gives permissions to b. a Matrix bot to manage creation of new rooms c. UI extensions to show this alongside communities.
However!
Lemmy is deceptively shiny and awesome, but there's still a lot of way more high-priority stuff that needs doing before this kind of huge feature extension is even discussed seriously, so the devs need to focus on that kind of thing.
Once the Element devs get embedded rooms a bit further down the road, this seems like a really doable project for a motivated Lemmy user to try adding on, though!