Means "love", in dog words. Or as a verb means that you are going to pet, hug, cuddle, and play with your dog. Especially related to the dog's nostrils because they are so cute and rubbery and pokable. You can't help but twurk them.
You say to your dog "I twurk you!" This is the same as saying "I love you!" to a human.
You say to your dog "I'm going to twurk you so strongly!"