“I had nothing to do with that.”

Your parents found out that your sister has started smoking & got angry at her. You already knew she was a smoker. Your sister thinks that you told them, so now she's angry at you. But you weren't the one who told them, so you say:

I had nothing to do with that.

have (something / nothing) to do with (something)

This phrase means to be connected with, or be responsible for, something. In the example above, you are telling your sister that you weren't responsible for your parents finding out that she smokes.

"Have something to do with" is more often used to talk about your responsibility for bad things than for good ones, but can be used for both.