The green color is related to the placenta, and is expected to quickly fade. The puppy is already less green than it was at the time of birth.
An article in The Sun U.K. states that there are only two other known instances of green puppy births. In one case from Spain, two green puppies were born in a litter in 2014. One of the two died shortly after birth. The first known case was a puppy born in the U.K. in 2012. In all cases the green color faded away completely in a week or two.
Sutherland and her family named the puppy Forest, on account of his green fur. The green color is supposedly related to biliverdin, a green bile pigment from the placenta. The green hue has no known health effects, even though the Spanish green puppies happened to be born weaker than others in the litter.
From NTD Television.