Taken from https://stadia.google.com/controller/index_en_US.html
After switching your controller into Bluetooth mode, press and hold the “Y + Stadia” buttons for 2 seconds to enter pairing mode. The status light will flash orange.
Then, go to the device you want to play on and pair your controller in the settings. Once paired and connected, the status light on the controller will turn solid white.
Congrats! You’re ready to start playing. The next time you turn on the controller, it should automatically connect to the last paired device.
Not all Bluetooth devices are the same, so compatibility will vary. The Stadia Controller uses Bluetooth Low Energy connections, so some features, such as pass-through audio, aren’t possible wirelessly.
We’ve verified that the Stadia Controller works for gameplay with the list of supported devices. It hasn’t been tested with all Bluetooth device types, so it might not work with others.
Hold the Stadia button for 5+ seconds to turn the controller off.