To move a provisioned instance of Teleport Station and keep all the configuration, follow these steps:
Save the contents of C:\Users\..\TeleportStationService\config directory or equivalent on Linux
Un-provision the device on the old computer, or just uninstall the app and delete the config directory.
Install Teleport Station on the new computer (do not start the service, if it starts stop it).
Copy over the contents of \config directory from the old computer.
Start the Teleport Station service.
That is it the device will start and show as Active on the Teleport dashboard.
Note: You may have to adjust camera settings in the Teleport Dashboard of content save paths to match the new computer.