I have the same problem. It's dumb, but the organizers specifically say "mp4 files are not acceptable".
Installing QuickTime Player doesn't help. I made sure I have the latest version installed. There is still no H.264 option listed in the codecs when QuickTime is the selected format, nor is there any preset that helps. (Motion JPEG, which is another option the film festival allows, is also not there; and of course ProRes isn't there because it's Windows.)
I don't want to export an mp4 and just rename the extension to .mov, because that doesn't change the file container format. Sure, some programs will still be able to play it, but QuickTime and mp4 are two different containers; if someone can't play mp4, then they won't be able to play the resulting file.
My plan for now is to export as mp4, then use ffmpeg to make a QuickTime file with the same H.264 stream (and the same AAC audio stream) as the mp4 file. For example, if my exported file is Filename.mp4, I would type:
ffmpeg -i Filename.mp4 -c copy Filename.mov
This runs very quickly and doesn't lose quality, because it doesn't decode and re-encode the data; it's just copying the same compressed data into the new container.