sometimes it's on a little slip of paper which came with the board. sometimes you can get it looking at the bios of the board (not sure about this one). You could try going to the mfg web site to see if they have anything to help.

If you are in peril of voiding your warranty I'd strongly suggest NOT opening the case. It's not worth it.

Why would you need the serial number?
Your system 'should' come with sufficient documentation and/or driver disks for you to determine the card 'model' which is all you would need for a driver update, etc.

Also if you are new to messing with a computer's entrails, I'd leave it/them alone.

I've watched brand new systems frying on occasion before, like a freak short in a CD-rom taking out a motherboard...[cpu and ram survived, at least]...