You need to deinterlace your source before compressing. QT5 Pro has a "single field" option (in the Quality settings, I believe), but it's only good for playback, not exporting (when will this be fixed?).
If you've got Cleaner you're in luck; make sure the Deinterlace option is checked (Set it to Auto) and your video will turn out much nicer.
If you only have this problem in one scene my guess is that your camera settings were different when it was shot, causing the footage to be shot interlaced. Progressive scan is the way to go.
Good luck, Mike!
BTW, if anyone can deinterlace in QT5, please, share your wisdom.