Where exactly is QuickTime Player on OS X 10.8.5 Mountain Lion?
QuickTime Player should be located directly in /Applications/, that's its default location. QuickTime Player X is part of the OS installation, and Apple provides no separate downloads of the app.

Reinstalling the OS is of course a huge step just to get QuickTime back. If you have access to the Mountain Lion installer, you can use an app like Pacifist to extract QuickTime Player from it.

Mountain Lion, as it was a paid upgrade, also cannot simply be downloaded. You may have a copy of the installer on an usb drive or a dvd-r. Originally it was an App Store download, and when bought an app file called "Install OS X Mountain Lion" was placed in /Applications/. After the installation this file, which was more than 4 GB in size, was deleted. You can get it back from the Purchases tab in the App Store to re-download the file. If it shows as "Installed," you should try holding down the option key while clicking on the Purchases tab. You might also be able to locate the installer on a Time Machine backup in the /Applications/ folder.