I want to know how to export a 2 minute QuickTime movie in Cocoa.
On Mac OS X Apple provides the QTKit.framework which offers convenient access to QuickTime's core functionality.
As you didn't provide any details in your question, I assume you want to export a 2 minute long sequence of a longer QuickTime movie on Mac OS X:
The "Making an Edit" section of Apple's "Using the QTKit framework" explains how to trim an existing movie and export the result.