What do you mean by “output” from Final Cut Pro? FCP’s output is usually a file, often rendered at less than real-time.
If you’d like to send clips and sequences to QC through Syphon, it is probably possible if you do some coding and write an FxPlug that would render in realtime, but then it would only work while you were playing those clips/sequences in FCP, and if FCP decided to render in realtime (depends on system, codec, other variables).
FCP is an awkward choice for real-time video editing. Do you have to use it?