If you want a little peace and quiet (as in, no runners, cyclists or buskers), the six-acre Conservatory Garden serves as one of the park's official quiet zones. And it's beautiful, to boot: bursting with crabapple trees, meandering boxwood and, in the spring, lots of flowers. It's located at 105th St off Fifth Ave.
Otherwise, you can catch maximum calm (and max bird life) in all areas of the park just after dawn.