Can I transfer Apple Care on my current watch if I upgrade?

No. Apple Care+ for Apple Watch is tied to the device and cannot be transferred.

If it was paid in full and there's still coverage remaining, you can cancel it, and get a partial refund for the remainder, and use that to buy coverage for the new Watch.

If you were paying monthly for it, you can discontinue the coverage and stop payments, and then start Apple Care for the new Watch.

See here for more details: How to manage your AppleCare plan - Apple Support

Apple Care cannot be transferred to other devices, but it may be possible to request a refund for unused portion. Use the link How to manage your AppleCare plan - Apple Support