I'm not aware of an easy, no-homebrew way to get the serial number.
What you could do would be to use Wiimmfi to get the serial number.
If you are using a Wii (not a WiiU) you can use this tutorial to connect to Wiimmfi:
As soon as you've done that, send me a private message with the 16-digit console friend code of your Wii (you can see that in the Address Book in the Wii Menu) and I'll be able to tell you the serial number of your console.