No, it’s only green when there is an active connection with a client. But it will show a number of clients served in the last 24 hours in the popup or on the icon badge (unless it’s 0).
If you’re feeling adventurous, you can go to about:debugging#/runtime/this-firefox, find Snowflake, click “inspect” and search the console messages for “websocket-relay connected!”. If that’s the case, the extension has tried to connect to a client at least once.
You should also see a bunch of “Polling broker” messages, which means that the extension is at least trying to get clients.