The idiomatic phrase "not a peep" almost certainly originates from the sounds that baby birds (and in particular, baby chicks) make as nestlings. Consider for example, this instance from Henrietta Peck, "Fowl Literature," in the [Chicago Illinois] Prairie Farmer (July 15, 1871):
Twenty-one days at last dragged its weary length to an end, and in a state of great expectancy, I called upon Bessie and Josepha to count the little valuables which they had been able to produce from my choice eggs Not a peep to be heard about the nests. The eggs were too valuable to be examined as other eggs would have been, so I waited another day, and another, and another. Then I found one solitary chicken under each hen—both buff Cochins.
Examples of idiomatic use of "not a peep" go back further still. From an untitled item in the Joliet [Illinois] Signal (November 7, 1848) reprinted from the [Joliet] True Democrat:
The meeting adjourned and gave 3 hearty cheers, that made the Court house ring again, for old Zach. The barnburners then raised a faint cheer for Van Buren, but not a peep could be raised for Cass.
From "Political News" in the Cleveland [Ohio] Morning Leader (November 21, 1860), quoting a reporter for the New York Evening Post who is writing from Washington, D.C.:
Many of the office-holders here are reckoning upon a continuance in office, and especially those who have friends in democratic Senators. They claim that Mr. Lincoln's cabinet officer cannot be confirmed without the consent of their senatorial friends, and the price of that consent will be the retention of certain democratic clerks in office. A good deal of speculation and bargaining—always on paper of—this kind is going on at the Departments, and not a peep of treason can be heard in any of them. Even the South Carolina clerks say they will remain in office just to oblige Mr. Lincoln. They are not afraid of South Carolina terrorism in Washington.
And from "The News" in the Cleveland [Ohio] Daily Leader (November 29, 1865):
Thirty-two women on a railroad train from Canada to the United States, the other day, had babies. A shrewd government detective on board observed that they were the quietest little bundles of juvenile humanity he ever saw, as not a peep was heard from them on the journey. This led him to investigate the baby question, which resulted in the discovery that what were to be little dears, were tin cases containing from five to ten gallons of whisky each. An ingenious method of smuggling, truly.
I've been familiar with the expression "not a peep" from a fairly young age, having run afoul of it as a third-grader. Our teacher had lined us up in a double row prior to walking to the school library and had appointed me as a "hall monitor" to help keep order in the ranks. She then told the class, "We must walk quietly because students in other classrooms are still at work. Remember: not a peep from anyone." Instantly it struck me that it would be immensely droll to say "Peep!" in as deep a voice as is possible for a nine-year-old to manage. So I did—and was instantly stripped of my special rank and relegated to the back of the line, where I could be kept under special observation during the brief march to the library. The lesson I learned: in comedy, timing is everything.