Not at all. All costs should be covered by the gift giver. They've taken care of everything - good on them.
The only times you'll be asked to enter payment information is if you add more bottles on top of your gift amount, change to a different price point or want to extend your subscription after your gift period ends.
Apart from that, sometimes we do get people accidentally adding bottles on top of their selection rather than replacing bottles. In this case, it may ask for your payment details, but please just remove any extra bottles or "to be recommended" bottles from your box so that you get back to your gift amount and you'll be good to go.