Yes, you can. When you buy a Gold membership via the Wizarding World website, you will be given the option to purchase it as a gift. During the purchase you will be prompted to enter the email address of your recipient. Once purchased, they will receive an email with their Gold membership code which can be redeemed online.
You can also purchase a Gold Gift Box in-store at the shop at Platform 9¾ Kings Cross, the shop at Warner Bros. Studio Tour London, in selected Barnes and Noble stores and via the Barnes and Noble website.