The Stadia controller should work wirelessly with devices with a Bluetooth Low Energy adapter. Google specifically tested it on Windows 10, Windows 11, macOS, Chrome OS, and Android, but I also managed to connect it to my Apple TV 4K without any issues.

How do I pair my controller using Bluetooth? After switching your controller into Bluetooth mode, press and hold the “Y + Stadia” buttons for 2 seconds to enter pairing mode.

begin Bluetooth pairing by pressing and holding both the 'Y + Stadia' buttons for two seconds (the status light will flash orange) then go to the device you want to play on to pair it with your controller in the settings (once paired/connected, the status light will turn solid white)

If your Stadia Controller won't connect to a device, try the following steps before you perform a hardware reset. Use a functional charger and USB-C® cable to plug your Stadia Controller into a power source for 30 minutes. release the button. for 2 seconds or plug it into a power source.