With Qt instead of registering callbacks you're deriving from QGLWidget and override the mouse and keyboard event handler functions (inherited from QWidget). The changes to your input event handling are minimal.
QGLWidget::resizeGL is much like the reshape callback and QGLWidget::paintGL much like the display callback.
The problem is that I have just discovered that QT uses it own OpenGl calls
They're provided as a convenience. You can use GLEW and your regular OpenGL calls just fine with Qt. Most importantly you must make sure to include glew.h before any Qt header in all relevant compilation units and the headers those use.