Internet Plugins are shared across browsers, so if an embedded QT movie works in Firefox, then the component is OK for all browsers. Files sent to an external app do not involve plugins at all.
When you doubleclick a QT .mov file, what application does it open in? Safari uses the Get Info default apps to open downloaded files. You may have set QT files to open with something else.
Alternative: corrupted Safari preferences. Try to duplicate the problem in a fresh test account.