Found this dog after getting revered with The Tillers. I don't know if you can get this quest earlier, but as I've been farming those Dark Soil items in the Heartland area for atleast an hour over several days prior to being revered, and never saw this dog, that's why I think you might need to be revered to see him.
After I had turned in the quest, the dog despawned(or entered another phase).
Edit: When I returned to my/Farmer Yoon's farm at Halfhill after completing this quest, there's a dog next to him just called "Dog".
If you do the /love emote to it, you will see the text "Your dog loves you too!", and if you do /pet you will see the text "Your dog pushes its head under your hand."
Maybe some other emotes will give a response too - I didn't try :)
The dog will also sometimes dig below the house on your farm... I'm thinking this dog might be needed to find that old map Fish Fellreed is talking about?
Edit: Ok; so the dog is not connected to the old map it seems.