I do not think this can be done with Qt. Blurring can be done using https://doc.qt.io/qt-5/qgraphicsblureffect.html but it is only limited to the widgets painted by Qt. Which the underlying background is not, even if you manage to make your widgets transparent or semitransparent. Painting the background is always the business of the operating system (or window manager) and not the business of your application Qt.
You can certainly try to do some extreme hacking like grabbing the active screen before your window is displayed (see How to capture the current screen image using Qt?) then getting certain rectangle content of the image, which corresponds to the background of your window, then paint it blurred to the background of your application and then update it everytime you move or resize your window... But anyway, even if you manage this, this background will be static and not dynamic.
I recommend that you abandon the idea of blurry background and leave this function to the window manager and the operating system.