You can initiate the Easter egg by clicking on the trees on the screen. Once the flying duck appears, you've only got five shots to hit it. If you miss, the "dog" — which, in this case, appears to be a Treasure Goblin from Diablo 3 — will pop up and laugh at your incompetence. The hunting minigame even includes Duck Hunt-appropriate 8-bit sound effects.
HeroesNexus recorded footage of the Easter egg; you can watch it above. For more on Heroes of the Storm, check out our review.