There seems to be two possibilities for the serial number
1) on the back of the controller directly under the handle is a sticker with a QR-code and number printed on it
2) in the DJI Go app, select the 3 dots menu '...' on top right corner, then again select the 3 dots menu '...' then at the bottom of that menu, select about. It shows the
P4p serial number, and the remote controller serial number.
I notice the serial numbers for either
P4p or RC do not show until the
P4p is powered on so this may not be easy if your'
P4p is not running.
Also, my sticker number on the back of my remote is not the same number as the serial in the menu so the sticker may or may not be any use - I did notice the stickers on both my controller and a friends' are different numbers so maybe DJI can use it if you cannot get the DJI Go to show you the serial number there.