The four-line poem “A Wise Old Owl” is cited in print from at least 1883, but is of unknown authorship. A 1904 source (below) credits “the immortal poet Bromley.”
 
The wise old owl (who either sits or lives in or on an oak) observes much, but doesn’t blab about it. The poem and its moral were used in World Wars I and II. The owl poem was a favorite of industrialist John D. Rockefeller (1839-1937) and Calvin Coolidge (1872-1933; president from 1923-1929).
 
   
Wikipedia: A Wise Old Owl
A Wise Old Owl is a nursery rhyme. The lyrics of A Wise Old Owl poem are derived from the saying ‘a wise old owl’ based on an owl’s behavior of watching and patiently waiting when hunting its prey.
 
Lyrics
A wise old owl lived in an oak
The more he saw the less he spoke
The less he spoke the more he heard.
Why can’t we all be like that wise old bird?
 
Other versions
During World War II, the United States army used the rhyme and tweaked the ending to “Soldier…. be like that old bird!” with the caption “Silence means security.”
 
21 July 1883, Rio Grande Republican (Las Cruces, NM), pg. 4, col. 2:
There was an owl sat in an oak; the more he heard the less he spoke; the less he spoke the more he heard; Oh, that men were all like this wise bird! 
 
Google Books
The Hoot of the Owl
By H. H. Behr, M.D.
San Francisco, CA: A. M. Robertson
1904
Pg. 11:
Owing to the weakness of my bill, I am a bird of few words; as the immortal poet Bromley sings:
“There was an owl that lived in an oak,
The more he heard the less he spoke,
The less he spoke the more he heard,
Oh, let us be like this wise bird.”
 
18 May 1909, New York (NY) Times, pg. 18:
ROCKEFFLLER VERSE
QUOTED TO NURSES
His Jingle Figures in the Opening
of the Training School on
Blackwell’s Island.
ADVICE FOR THE INMATES.
William Rhinelander Stewart Gives
Them a Few Points on How to
Act in a Sickroom.
Joseph H. Choate told William Rhinelander Stewart at the opening exercises of the Metropolitan Training School for Nurses on Blackwell’s Island yesterday that he was the bravest man he had ever seen. This was after Mr. Stewart, who as President of the State Board of Charities had made an address, gave the nurses who are occupying the new building some good advice.  He first quoted to them this little rhyme, one of the articles of faith proposed to his followers by a man who Mrs. Stewart admitted later was John D. Rockefeller:
 
A wise old owl sat on an oak;
The more he saw the less he spoke;
The more he spoke the less he heard,
Why can’t you be like that wise old bird?
   
8 August 1909, Springfield (MA) Republican, pg. 7 ad:
“A wise old owl sat on an oak,
The more he saw the less he spoke,
The less he spoke the more he heard,
Why can’t we all be like this bird?”
Get Wise and Use
MASSASOIT COFFEE
 
20 August 1909, Springfield (MA) Republican, pg. 1 ad:
A wise old owl sat on an oak,
The longer he stayed the less he spoke,
The less he spoke the more he heard;
Why are not more of us like that wise old bird?
SAY NOTHING,
BUT INSURE WUTH
OPPENHEIMER & FIELD
 
Chronicling America
8 September 1909, Washington (DC) Herald, pg. 7, col. 2:
“A wise old owl lived in an oak.
The more he heard the less he spoke;
The less he spoke the more he heard.
Why aren’t we all like that old bird?”
 
Chronicling America
12 June 1910, Washington (DC) Herald, pg. 1, col. 2:
A YOUNG attorney of Washington was invited to deliver an address to a graduating class of trained nurses at one of the city’s hospitals. It was suggested to him that he might make a hit by giving the young women a motto to guide them in their work. This is the motto he sprung on them:
A wise old owl sat on an oak.
The more he heard the less he spoke.
The less he spoke the more he heard—
Let’s imitate this wise old bird.
 
4 January 1911, Aberdeen (SD) Daily News, pg. 2:
O. W. Coursey in Sioux Fallas Argus-Leader: The Quakers of old Pennsylvania were noted for their silence, particularly so in their moder of worship, where not a sound was heard, and each penitent prayed to himself. THey ha a verse which they used to teach to their children. It ran as follows:
“A wise old owl set on an oak;
The more he saw, the less he spoke;
The less he spoke, the more he heard,
Let us mimic that wise old bird.”
   
Google Books
January 1912, Our Work (Bulfinch Place Church), pg. 4:
A Wise Old Owl
“A wise old owl lived in an oak;
The more he heard the less he spoke,
The less he spoke the more he heard,
Why aren’t we all more like that bird?”
 
20 September 1914, New York (NY) Times, “Appeals to Readers,” pg. BR394:
WAPAKONETA—Can any of your readers give me the author of these lines:
 
A wise old owl sat in an oak,
The more he saw the less he spoke,
The less he spoke the more he heard;
Why can’t we be like that wise old bird?
 
18 March 1917, Aberdeen (SD) American, section 2, pg. 11:
A wise old owl lived in an oak.
The more he saw the less he spoke;
The less he spoke the more he heard.
Soldiers should heed the wise old bird.
 
17 March 1918, New York (NY) Times, “Appeals to Readers,” pg. 80:
“Wise Old Owl”
B. A. WILLITS.—Can any reader tell me where I can find the following lines:
A wise old owl lived in an oak.
The more he saw, the less he spoke.
The less he spoke, the more he heard.
Why can’t we be like that old bird?
 
25 Auguswt 1918, New York (NY) Times, “Answers from Readers,” pg. 46:
The Wise Old Owl
MISS MARIE J. FAST.—Inanswer to the question asked by B. A. Willis in THE BOOK REVIEW as th=o where he can find the lines of the “WIse Old Owl”:
A wise old owl lived in an oak.
The more he heard the less he spoke.
The less he spoke the more he heard.
Why can’t we be like that old bird?
Would say that the lines are from “Poor Richard’s Almanac,” by Benjamin Franklin.
 
24 August 1923, New York (NY) Times, pg. 10:
A reader o THE TIMES says that in all that has been printed about President COOLIDGE of late he has seen no mention of a poem that hung on the wall back of his chair while he was Governor of Massachusetts. Whether the verses were put up in his home or his office in the State House the correspondent has forgotten, but thus, he says, ran the lines:
 
A wise old owl sat on an oak.
The more he saw, the less he spoke.
The less he spoke, the more he heard.
I’m going to be like that wise old bird.