No. The Queen has no role in Government. The G-G fulfills her constitutional duties. And all other constitutional powers such as the power to choose the G-G or to declare war etc. has been passed onto the Prime Minister.

The Crown does apparently still have power constitutionally to disallow a bill passed by Parliament within one year of that bill being given assent by the G-G. This power has never been exercised - and if it were attempted, I think a referendum would remove that power very quickly.