Yes, but it's not a country. The only country in the world that forbids eating dog meat is Hong Kong.

Chapter 167A-section 22 of the Dogs and Cats Ordinance

(1) No one shall slaughter any dog or cat for food, whether for human consumption or not.

(2) No one shall sell or use or allow others to sell or use dog meat and cat meat as food.

(3) A person who is found in possession of the carcass or any part of the carcass of any dog or cat is guilty of an offence under subsection (1) or (2) unless he satisfies a magistrate that the dog or cat was slaughtered, sold or used as food only in contravention of this section.

This is the only law in the world that prohibits eating cats and dogs.