What Brook converter are you using? They make over a dozen different types. The FGC2 converter used for the PS5 will not work with the Nintendo Switch or NS2. The NS Lite and XE 2 should work.
Most Brook converters will work with the Quadstick in PS4 emulation modes, which should give you motion and touchpad controls. Quadstick USB emulation modes that should work are 0, 1, 3, 4, 7. The Nintendo Switch mode 5 will not work however, so it you are using a config file meant for the NS1 with a Brook adapters, you will need to change the quadstick's emulation mode to something else. The same settings that work for the FGC2 should work for the NS Lite or XE 2.
Brook frequently updates their firmware. If your converter is not working with the NS2 it may be a firmware issue. You can find the latest firmware for your converter here:
https://www.brookaccessory.com/download/Wingman_Series/ Find the recommended firmware version number and update your converter. You'll need a Chrome or Brave browser. Edge won't work with their update page. They often have experimental versions of firmware. I generally use the one listed on their webpage.
All their converters will work with the PC as an XBox controller. You can troubleshoot the problem by connecting the converter to the PC, connect the Quadstick to its input and check for it working using joy.cpl or a test page like:
https://test.quadstick.com. If the Quadstick controls the outputs of the converter on the PC, then we know the problem is between the converter and the NS2, or a setting in the NS2.
The converters emulate a USB Nintendo Switch Pro controller, so the Switch System Settings->Controllers and Sensors will need to have Pro Controller Wired Communication turned on. It is off by default in new Switches.
What are the lights on the sides of the converter doing? They should be a steady Blue. If they are blinking, then the converter is not communicating with either the console or the quadstick.