The first Nintendo game I really loved was Donkey Kong Jr in the arcades. However, I had no idea what Nintendo was back then, so it definitely didn't make me a Nintendo fan. Nintendo was not a household name in the early 80s.
If I had to pick one game that made me actually love Nintendo, it would be The Legend of Zelda (NES). I had played plenty of console games before Legend of Zelda, but I always felt like I was getting an inferior experience. The arcade games were always better than the console games. I had also played some computer RPGs at my friends' houses like Ultima and Wizardry, and these were great for a different reason.
But then Zelda comes along, and that is when I saw that console games could be better than either arcade or computer games. The Legend of Zelda had all of the great action of an arcade game and the depth of a computer RPG. It was the best of both worlds all in one game. This game is what made me think of consoles as the premium platform instead of an inferior platform. And at the time, Nintendo was the king of console gaming. Zelda is the game that made me a Nintendo fan.