Sadly, they just closed/removed the best quiet spots - on TSI.
Leaving the park for one of the monorail resorts is a nice option. The monorail resorts have lots of quiet spots to relax, but taking the boat is more relaxing/quiet than taking the monorail to get to them. The smaller boats are probably the most quiet/relaxing.
Within the park, there's a quiet area over by the Space Mountain restrooms. Peoplemover is also kinda quiet. There is, or was, a quiet spot by the pirates' restrooms, but the construction might have changed that, or it might be blocked. There is also a sorta quiet spot over by Tiana's, again by the restrooms over that way. There is also a semi-quiet spot by the B and B restrooms.
But you specifically mention August. Taking an afternoon break out of the heat is often key to enjoying WDW in August. Plus, there are often pop-up downpour storms. An indoor location with air-conditioning is helpful that time of year, so that brings me back to suggesting one of the monorail lobbies, or Wilderness Lodge. If no conventions, CR has a number of places to relax. GC lobby isn't quiet, but the piano music can be relaxing. Poly has a few options, too.