I searched high and low for an option to disable the camera shutter noise, but as far as I can tell, there isn't one. After a bit of searching, I pieced together a solution. Please note that this is not a supported method of working on your phone and messing with the registry is a good way to brick your device. Play around with this at your own risk.

1. Download PHM Registry Editor. I chose the desktop installer.
2. Run the installer, but don't worry when it doesn't actually install to your Q.
3. Browse to the folder on your PC where the installer dumped the CAB files. Manually them all to your device. I think you only need "regedit.Stngr_ARM.CAB" but they're small so I just copied them all.
4. Open the File Manager on the Q, browse to the folder with the CAB files and run regedit.Stngr_ARM.CAB. After it installs you'll get a warning about the app not working because it was made for an earlier version... ignore that. You can now delete all the files you copied to the device.
5. From the home screen on the Q, click Start. Start PHM Registry Editor.
6. Navigate to \HKLM\System\Pictures\Camera\OEM
7. Click the "Values" left soft button.
8. Select SoundFile
9. Change the string from "\windows\shuttersound_02_secs.wav" to "\windows\*none*\" (asterisks are required.)

Thanks to the guys at www.everythingq.com for pointing me in the right direction.