Can I find my lost Apple Gift Card code with the serial number?

If it was a physical card, then sorry but no, probably not. Gift cards are generally like cash; if you've lost it, it's almost certainly gone. But you can contact iTunes Support and ask.

Choose your country or region - Official Apple Support

Regards.

Totally different issue and should have been posted as a new thread (or actually just been searched for as it's been answered dozens of times) but:

If you can't redeem your Apple Gift Card or App Store & iTunes Gift Card - Apple Support

I have apple gift card here $500 but I miss some code