brooklyn art museum
which one? the Brooklyn Museum?
Barboncino Pizza is really good; I'm not sure if it's especially QUIET, but depending on where you sit it can be, I guess. There are several good options along Franklin Street nowadays, in fact.
You could also strike into Park Slope, where innumerable places along 7th ave are like you describe. Scottadito, for example, or Hunan Delight on Union Street.