Ubuntu used to have an option for that and the screen shot from the site looks quite a bit like the old Ubuntu installer.

Oh, right you are. https://help.ubuntu.com/community/Wubi

It even has a wikipedia page! https://en.wikipedia.org/wiki/Wubi_(software)

From the documentation site you linked, Q4OS doesn’t run inside Windows. The Q4OS installer runs inside Windows, and it stores the root filesystem for Q4OS as a file inside an existing Windows drive partition. I’m not personally aware of other Linux OS’s that do either of these things, nor do I particularly desire a Linux OS that does either of these things. I can see it being useful if you want to keep your Windows installation fully intact. I imagine disk I/O performance takes a small hit.