Quad Cortex does not work like HX Stomp in this regard. Like Peter said, no midi messages are sent when you press buttons on QC.
To do what you describe, you'll need some other device programmed to send the midi messages that your QC will respond to (just for programming). This can be a device like a Morning Star controller or an app, like Midi Studio or Midi Designer Pro X on your iPad.
Make sure you're familiar with the MIDI capabilities of the QC. For instance to engage Scene A: CC#43
Configure the app or device to send CC messages for each of the Scene changes, you may also wish to start each song by switching to Gig View and Scene Mode. Other techniques might be to switch to Stomp Mode, Press Button B, then switch back to Scene Mode. All of these things are possible to do rapidly and consistently.
Once you've programmed the various options, use the app/device to send the midi messages to ST4 in Lyrics mode while it listens for incoming MIDI. You'll see the messages written out in the lyrics editor as text.
Once that's done, connect ST4 to QC MIDI (via USB for instance) and run the playback to test.