No, people do not know that Converse is part of Nike. I do not believe this is common knowledge. I did not know that Converse was part of Nike until I saw a headline stating such today. I suspect that many people are also unaware of this, and would need to be told.
I always thought they were competitors and there are certainly very different fans of each brand. This is probably a good way of diversifying in the same market without cannibalizing your marketshare. That said, I am not into either of these brands, so it doesn't really make a difference to me.
I certainly did not know that Converse was a part of the Nike company before reading this headline, and I suspect that most people are also unaware of this fact. There is no real reason why people should know which company Converse is a part of, unless they are trying to avoid spending money on Nike products because of their child labor scandals.
People aren't very aware of the specifics of brands, which is ironic, considering how much time and energy brands devote to making themselves known. At the same time, it probably doesn't matter too much that people don't know that Converse is a part of Nike. If people needed to know, they'd just call it Nike instead of Converse!