Q: Which chicken breed has the quietest, tamest hens??
A: Hens of most breeds are rather quiet except for right before and right after they have laid an egg. Then they may make a loud "announcement" about it for a few minutes, usually called a cackle. They seem to be very proud of their accomplishment! Some breeds like Brahmas, Australorps, Faverolles and Mille Fleur D'Uccles are often specifically described as being quiet, but frankly I've never seen too awful much difference between a "quiet" breed and a "normal" breed in my flock. Hens of most breeds are fairly unobtrusive.
Antwerp Belgian Bantam hen
Young pullets who have just begun to lay may be louder about it for longer--at least for a while until they get into the swing of things. We think the process of laying is surprising and uncomfortable to them, and it takes a while for them to properly interpret what their instincts are telling them to do. Chances are good that your hens will cease making a long to-do once they get used to the laying process.
Additionally, hens are not loud once the sun has gone down, because they're sleeping.
Of course, it's always possible to get a loud individual (as individual personalities vary), but you can probably assure your neighbors that your girls just are gearing up for the season to be able to provide them with tasty eggs, and that after they've had time to adjust they will probably only speak loudly for a minute or two before or after laying. "Loud" hens are still quieter than dogs.
To our knowledge, there is no way to ask a hen (or a rooster) to be quieter. Whenever we ask ours to shush, they act as if they can't understand you!