I like to create an application that runs in real-time.
You can not run (even though you somehow manage create it) an application in real time, on a non-real time machine.

It needs to run at a very high frame rate.
My question is, should I therefore build my application as a qt console application or should I build it like a normal widget QT application with basic window functions?
Anyone has experience if this choice between console and non-console makes a difference?
You say high frame rate, does that imply you need a GUI.

What I think is console or GUI should be decided based on use interface requirements, one can write GUI with high performing back end (non-GUI) components. It is also possibly to write a low performance console based application.

I would say to go with a light weight GUI, and high performing components in back end and always exploit the components provided by frameworks.