The Qutebrowser dependency list says it can use either QtWebEngine (Chromium) or QtWebKit (WebKit):
> QtWebEngine, or alternatively QtWebKit (5.212) - This is not recommended due to known security issues in QtWebKit, you most likely want to use qutebrowser with the default QtWebEngine backend (based on Chromium) instead. Quoting the QtWebKit releases page: [The latest QtWebKit] release is based on [an] old WebKit revision with known unpatched vulnerabilities. Please use it carefully and avoid visiting untrusted websites and using it for transmission of sensitive data.
To use Geko (or hopefully Quantum) with Qutebrowser you would need a nice QT integration. Poking around a bit, it looks like Mozilla and Nokia did some work on this in 2008, but it doesn’t seem it ever got out of beta: