The answer is no. The last OS it'll work on is Sierra or maybe High Sierra. And, even if you have that OS version, you won't be able to install it because the installer is not universal (the app is though), Mine is still good on the iMac I keep on the old OS version because many years ago, I had a copy of an Apple article on how to uninstall FCE and kept it, so I would manually copy every file, create a new folder on the new Mac for it, and spend about 2 hours on moving all of them (and there were a lot), avoiding having to use the installer. It worked, but if I choose to update the iMac, it will be lost.
Having said all that, I have no idea if it would work on an M1 - my guess is that it would not.