The "Visit College of Winterhold" is still listed as an unfinished misc quest in my journal, even though I already discovered the college and also completed "The First Lesson" quest.
The active quest target is pointed to an elf named Nirya. However, when I tried to talk to her, nothing happen, the target arrow is still pointed on her head.

Is there a specific way I am required to do to complete this quest?

I'm on version 1.3 by the way.

Did you show her your magic or shouts?

Yes, I did. That why Faralda let me in to the college.
But the active quest target is pointed to another she-elf, Nirya, that always say that college need some changes in leadership, and seem paranoid about someone spread lie about her.

According to http://www.youtube.com/watch?feature=player_detailpage&v=vuPgZLu82dU#t=267s, this misc quest should be completed when you first talk to Faralda (don't even have to cast a spell).
So, this is obviously a bug that the game points to Nirya as the quest target instead of Faralda.

Same problem here. I've completed the quests as well but it still shows. I've also completed the Tour so I now have First Lessons in my quest log. I've also completed several other quests that I picked up at the college but the Visit quest remains.

I've learned to ignore it since it doesn't seem to have any affect on progressing through the rest of the college quest lines.