Earlier quoted context omitted.
I haven't investigated Qt's mobile offerings or "Qt Quick", but I am fairly certain that on X11-based systems the basic core desktop SDK is as native as any other UI toolkit (e.g. GTK), and on Windows it uses the same core libraries that MSVC-written programs use, so it's as close to "native" as it gets there, too. Is that incorrect?
Little bit of a, little bit of b. They use some native controls on X11, and I think they hook into the Windows C++ APIs in places as well, but with the shift to QML I think it's back to being heavily custom control based again.
They are native controls on X11. Or, more accurately, X11 has no native controls. The closest is probably XAW, which is so awful it was dropped decades ago. There are QT themes that will use GTK for rendering, though.
(For reference, a screenshot of XAW: http://www.iue.tuwien.ac.at/phd/halama/_13595_figure4289.gif The box with 'Athena Simple Widgets' is what is provided by X11.)