There are several good "quiet" spots in IOA:
Port of Entry: go straight ahead, towards the lagoon. It's a very picturesque area that is ignored by most. The most noise is from the Hulk's screamers
Marvel: Spot 1-- veer to the right and go on the lagoon side. This puts you "behind" the Alterniverse store and is a HUGE area. It used to be the best fireworks viewing area. Now it's just a quiet area, sans Hulk screamers.
Spot 2--directly across from the restrooms, next to the ice cream stand. It's a small area with stone benches, but is quiet.
Toon Lagoon: Spot 1--on the left, by the amphitheater. If there is not a show going on, the whole area is very quiet. The only noise is Ripsaw screamers and games attendants.
Spot 2--along the far right side, where the lagoon is. The best entrance for this is near Popeye's ship. There is a long path with all sorts of cool themed props. You'll hear screamers from Popeye and Hulk at most.
Jurassic Park: just the huge area on the lagoon side of the Discovery Center. It's pretty quiet overall
Lost Continent: along the lagoon side, near Mythos. This area is pretty quiet and easy to miss.
Seuss Landing: along the lagoon, specifically Sneetch Beach. That area is behind Green Eggs & Ham
At USF there are some quiet spots. Some are in smoking sections, so take that as you will...
In front of the concert stage, near Rockit. As long as no concerts are planned, this is a very popular "rest" area. People are often seen taking naps on the "grass". Some screams from Rockit riders will be heard.
The smoking section across from Twister. It's a quiet area overall.
The alleyways in New York. Many people ignore these areas, so it's a great quiet spot.
Near the restrooms across from Louie's (Mummy side). Also a smoking area, but very quiet overall.
Along the lagoon side of Lombard's Landing and Richter's. Very quiet spot.
Across from The Simpsons ride entrance, there is a sitting area that is also quiet.
Anywhere along the lagoon, especially in the grassy walking paths running on the Kidzone/Hollywood/Simpsons side.
The grassy "park" area near Mel's. May also be a smoking area.
The Gardens of Allah area between Kidzone and Horror Makeup Show. It's a nice shortcut and virtually unused. Super quiet area.