Use the gear in StreamDeck configuration to backup your config.
If you don’t, the files are strewn about, so if you’re doing a selective restore, you’ll need to hunt down:
/Volumes/backup/Users/you/Library/Application Support/com.elgato.StreamDeck
/Volumes/backup/Users/you/Library/HTTPStorages/com.elgato.StreamDeck
/Volumes/backup/Users/you/Library/Application Support/elgato
- Perhaps others (the above worked for me)