According to the information I found, you cannot start an Apple Care subscription with a gift card. You must add an active payment option. Then the monthly payments can be deducted from your gift card. However, you must remember to redeem your gift card.
"For some recurring charges, like AppleCare+, you can't start your subscription with Apple Account balance. But as long as you keep another valid payment method on file and your Apple Account balance would cover an entire charge, your recurring charge will be pulled from your Apple Account balance."
You can find more information here:
What you can buy with your Apple Gift Card or Apple Account balance - Apple Support