What happens if I fart on someone’s balls

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.

The 4 people you have to fart on are:

1. Mayor of South Park. (Located in her office)
2. Father Maxi. (Located on a bench outside of the Mayor Office)
3. Principal Victoria. (Located in the community center)
4. Minister of Montreal (Locked in a cave in Cananda. You need to fart on him before talking to him or he disappears forever).

This achievement might be glitchy. I for sure had farted on everyone, but after farting on the Minister of Montreal, I had to take another round of farting on the South Park Officials. I had the achievement unlock after using the 'cup a spell' fart on everyone.