That's an interesting idea, to run Hyper-V completely without Windows. I think it's not possible, at least not without some substantial amount of hacking.

But it's no problem to run Linux on Hyper-V. It's a hypervisor, off course you can start nearly any operating system as a VM. You can also give the VM access to some hardware components. But I don't think it's possible to get a full native Linux desktop experience, with GPU/Screen, Keyboard and Mouse connected to the host system.

Edit: this post seems to answer your question, not sure if it's correct: https://superuser.com/a/1531799