Did you elect to make your QuickTime file self-contained when you created it? Obvious, but I had to ask. If you created a reference-only QuickTime file, it would only work on the computer you created it on.
I transfer files between my PC at work and my Mac at home via a USB thumb drive all the time. I plug it into my PC at work and it shows up in Windows Explorer. I plug it into my Mac and it shows up on the desktop and in Finder.
Never needed any additional software on either system.