"Blow" (with this meaning) is not a noun. It is only a verb. "The fan blows (air) on Peter."
Flowing air is called "a breeze". A breeze may be from a fan or air conditioner indoors, or from the wind outdoors.
You just said what I am trying to say. ‘Peter is blowing air on himself with the fan.’ ‘Peter is enjoying a breeze from a fan’ is also a sentence that I am looking for.Yes, you can say "the fan is blowing (the) air on Peter" but it's not especially typical of the way a native speaker would say it. It's better than "the blow of the fan" though. I'm not sure how to say what you seem to be trying to say, exactly. We'd be more likely to say that Peter is blowing air on himself with the fan than that the fan is doing it, since it's hand held, I think.
"Blowing (air) on Peter" doesn't mean Peter is holding the fan. It means the moving air hits Peter. If it is an electric fan, it may be on a desk or shelf or on the floor. It may be a fan someone else is holding.So for my situation, we can say ‘the fan is blowing (the) air on Peter’ because Peter is holding the fan?