.MOV's in and of themselves shouldn't be bad quality and just switching to a different file type may not fix your problem. What resolution, bitrate, framerate, and codec are you uploading at?
I just bought a Nikon dslr and am starting to use that. Up to this point, I've just used my iPhone 5's rear camera. I know it does 1080p videos and I think I usually have 25 or 30 fps. The quality is always good when I'm editing and when it's in an mp4 format, but when I convert it to quicktime, the videos become really grainy. I'm curious as to what is happening because I don't want the same thing to happen when I'm using my Nikon.
I just exported out my recent video in Quicktime and there's no noticeable difference. I think the color may be slightly different but everything was fine. So, it may have something to do with how you're exporting. But first, are you recording your videos and then converting them to Quicktime before editing? If so, that's probably the issue. If not, then it's probably the way you're exporting the video.