If you are running a Mac, then QuickTime (.mov) is installed automaticaly as part of Mac OS. HitFilm will use it to provide the best playback performance possible for .mov files.
On Windows, Quicktime is not required. HitFilm can read .mov files that use the ProRes codec, the Cineform codec, and the H.264 codec natively, without requiring Quicktime. If you wish to install Quicktime to support other .mov codecs, you can do so. But it is not necessary for general HitFilm use.