Ah! This seems to be an increasingly common request. I am not a Qt expert but I can point you to a recent post - OpenGL driver for windows. Does this help?
If you want to jump into the Geant4 source code, you’re on your own!! 
But all the Qt source code is in geant4/source/interfaces and visualization. If you make progress, it may help us to know what you have done so that we could offer this option in future.
John