If the iPod was the last device that you connected to and synced with your iTunes library, there is a chance you might be able to get the serial # to show up in the Device Connectivity diagnostic report. See here for instructions on running this test.
If you have previously registered the device with Apple, you can view it's serial # in your list of devices you have registered under your Apple ID.
See here: iPod: How to find the serial number
If none of those help, I'm afraid you are out of luck.