Contrary to the title, you should be looking for a wolf rather than a dog. As you saw yourself, there aren't many dogs to be found, if any at all.
To find a wolf, go to the lost forest, take a few turns, and you should easily find yourself a wolf to fart on.
A common issue with this is that if you progressed to the late game, after the zombie outbreak (which you have), the wolves in the forest will be replaced with the Nazi zombies. To fix this, you should finish the game storyline, which will bring back the wolves.
It is recommended that you use Cup A Spell for this to keep yourself out of combat, and make sure that after you fart on a wolf you maintain your gnome form until achievement completes, since it can be delayed.
Wolves in Canada won't work for the achievement.
Guess the Canadian medical insurance covers getting farted upon.