I am not a lawyer, but it seems like exporting Quickdraw Contour data might be prohibited by the end-user license agreement:
Export Control. You agree not to export from anywhere any part of the Application or any direct product thereof except in compliance with and with all licenses and approvals required under, applicable export laws, rules and regulations.
It might be possible to load the data into BaseCamp, a free GIS software from Garmin. BaseCamp has some (limited) export options:
Overlay data, including Garmin Custom Maps and BirdsEye imagery, cannot be exported to a file. To export overlay data, you must back up your data. -BaseCamp user manual, "Exporting Data"
You can back up all data, lists, and folders to a single archive file. The archive file can be restored to a different BaseCamp™ installation. -BaseCamp user manual, "Backing Up Data"
If the data you want to export is data you collected yourself, it does seem like you should be able to to what you want with it. I would recommend contacting Garmin customer service about it.
Just Googling "convert svy file" and "convert qdc file" does yield some results that claim to convert these file types, but they don't look especially reliable or safe.