Right now the Qt world doesn't have an acceptable browser, which is incredibly annoying for us since we're targeting Qt-based desktop environments. I'm hoping the move to GTK3 at the very least helps towards that.
As I understand it, even during the Nokia days, the point was to use Gecko with a Qt-native UI, so it wasn't about getting the entirety of XUL-based Firefox polished on Qt.
Ubuntu (mainly phone, coming soon to desktop) has these:
https://launchpad.net/webbrowser-app https://launchpad.net/oxide
It's based on Chromium/Blink.
And Qt itself switched from QtWebkit to Chromium with QtWebEngine - https://wiki.qt.io/QtWebEngine
From what I can tell the progress is backwards. A few years ago there were Firefox Qt builds on mozilla and regular progress on it. Now the builds disappeared and the only commits to the Qt backend are in some hard to find github repos with no activity.
Really the main visible difference is the file/print dialogs. I think maybe XDG would be a better way to standardize that, and would fix many other apps as well.
What project are you working on?
Qupzilla-git i.e. the version based on Qtwebengine is getting there, so reasonably soon we may have a decent Qt-based browser (I don't doubt scrollaway knows that). Also the fiber browser looks promising. I do think we need a way to do CI between chromium and Qtwebengine, or else we'll end up shipping browsers lagging one or two versions behind chromium.
As far as gecko goes, I think it's over. The hope is servo, which being much more modular may be integrated without struggling with XUL.