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Often used as a direction, the phrase birds in their little nests agree, and its variants, mean: children, especially siblings, should not argue among themselves.

This phrase occurs, for example, in the following poem from Hilaire Belloc’s Cautionary Verses (New York: Alfred A. Knopf, 1941), by the French-born British author Hilaire Belloc (1870-1953) [page 144]—here, the author humorously refers to bird’s nest soup, which designates a soup that is considered a delicacy in China, made with pieces of the nests of certain species of swiftlet native to Southeast Asia:

On Food

Alas! What various tastes in food,
Divide the human brotherhood!
Birds in their little nests agree
With Chinamen, but not with me.
Colonials like their oysters hot,
Their omelettes heavy—I do not.
The French are fond of slugs and frogs,
The Siamese eat puppy-dogs.
[&c.]

The phrase birds in their little nests agree and variants allude to the following piece from Divine Songs Attempted in Easy Language for the Use of Children (London: Printed for M. Lawrence, 1715), by the English hymn-writer Isaac Watts (1674-1748) [page 25]:

SONG XVII.
Love between Brothers and Sisters.

I.

What ever Brawls are in the Street
There shou’d be Peace at home;
Where Sisters dwell and Brothers meet
Quarrels shou’d never come.

II.

Birds in their little Nests agree;
And ’tis a shameful Sight,
When Children of one Family
Fall out, and chide, and fight.

These are, in chronological order, the earliest occurrences of the phrase birds in their little nests agree and variants that I have found:

1-: From The duty of loving and seeking Christ early recommended. An address, delivered in the Gaelic Chapel, Aberdeen, on the evening of 1^st January, 1818, to the children attending the Aberdeen gratis Sabbath evening schools (Aberdeen: Printed by D. Chalmers and Co., 1818), by the Rev. Duncan Grant [page 10]:

Children who love Jesus Christ, love their brothers, and sisters. You know that Cain was very wicked, and killed his brother; that Ishmael was cast out of his father’s house, because he mocked his brother; and that Joseph’s brethren, who hated him, and sold him, acted very wickedly; grieved the heart of their good old father, and displeased God very much. Christ tells you, that he who “says to his brother, Thou fool, shall be in danger of hell fire.” Whilst the little birds agree in their nests—whilst the gentle lambs lie down together in peace—should not children agree, and love one another?

2-: From On the Employment and Association of Birds, by J. R. Prior, published in The Monthly Magazine; Or, British Register (London, England) of March 1824 [page 111, column 2]—here, however, birds in their little nests agree is merely a quotation explicitly referring to Isaac Watts’ poem:

Twenty birds, at least, meet of a morning in a tree before my bed-room window. Their regular devotion awakes my heart and inspires my love to join in their early praise. When an hour has elapsed, these birds separate to their several avocations. Their instinct guides them to food and industrious habits. ‘Birds in their little nests agree,’ says Watts. Washington Irving has shown, in his ‘Tales of the Hall,’ at the ‘rookery,’ what birds can do, and how tenacious they are of propriety and decorum in all their household discipline.

3-: From The British Months; A Poem, in Twelve Parts (London: John W. Parker, 1835), by Richard Mant (1776-1848), Bishop of Down and Connor [Volume 1, February, page 51]:

Yes! ’tis a grateful sight to see
“Birds in their little nests agree:”
And grateful is the sound to hear
Their lively chirp, their warble clear,
Unprompted, unconstrain’d by art,
Spontaneous from the swelling heart!
But birds, like other things of earth,
Give symptoms of terrestrial birth:
And join with all that breathe to show,
That nought is perfect here below;
And, who would taste unmingled love,
Must quaff it from its Fount above.

4-: From Killed by Mistake, by Edward Mayhew, published in The Library of Fiction, or Family Story-Teller; Consisting of Original Tales, Essays, and Sketches of Character (London: Chapman and Hall, 1837) [Volume 2, page 209]:

Appearances are deceitful. The gentlest-looking creatures are often the most dangerous. The cat is a more terrible antagonist than the porcupine.
Once I thought differently. Once I formed a theory, in my young days of delightful ignorance, that man was the only animal having a natural love for the pugilistic.
The first line of that beautiful hymn, Birds in their little nests agree, suggested an easy and pleasant mode of testing my opinion, and I for a few pence procured a “little nest” of bullfinches. It was an interesting experiment when, with a mess of chopped egg and raw beef, I advanced to feed the emblems of concord with a skewer.
No sooner had I made my appearance, than the five half-naked creatures stretched forth their necks, each screaming at the top of its tiny voice, straining its jaws with unbecoming voracity, and, as if not satisfied with the choking size of the mouthfuls I administered, making more than once serious attempts to swallow the instrument on which they were presented. When I locked up my fledglings for the night, it was with a conviction that, at any rate, birds were not patterns for little children, as to behaviour at meals.
The next morning, however, on looking at them, I found the two largest and strongest of the family in possession of the nest; they having actually kicked their dear little brothers and sisters out of bed. There lay the two unnatural fledglings, comfortably snoozing, with half-closed eyes, no ways disturbed by the piteous cries of the ejected, who were tumbling about the bottom of the box in heart-rending disorder; nor was I able to arrange the difference between them. What they quarrelled about I cannot imagine; but, as often as I put the little ones to bed, the larger thumped them out again, so that in the end they caught cold and died. Their death-groans must have been heard by their unfeeling relatives, who, however, still seemed to doze as luxuriantly as usual, insensible even to the horrors of fratricide!

5-: From A Diary in America, with Remarks on its Institutions (Philadelphia: Carey & Hart, 1839), by the British author Frederick Marryat (1792-1848) [Volume 1, chapter 30, page 233]—the author is in St. Louis, Missouri:

Visited the museum. There were once five large alligators to be seen alive in this museum; but they are now all dead. One demands our sympathy, as there was something Roman in his fate. Unable to support such a life of confinement, and preferring death to the loss of liberty, he committed suicide by throwing himself out of a three-story-high window. He was taken up from the pavement the next morning; the vital spark had fled, as the papers say, and, I believe, his remains were decently interred.
The other four, never having been taught in their youth the hymn, “Birds in their little nests agree,” fought so desperately, that one by one they all died of their wounds. They were very large, being from seventeen to twenty-one feet long. One, as a memorial, remains preserved in the museum, and to make him look more poetical, he has a stuffed negro in his mouth.

6-: From Death and the Doctors. A Fable, an unsigned poem published in The Cheltenham Looker-On; A Note Book of Fashionable Sayings and Doings (Cheltenham, Gloucestershire, England) of Saturday 11^th February 1843 [page 94]:
—Context: Three doctors, a hydropathic doctor, a homeopathic doctor and a “doctor of the genuine school”, have vaunted each his favourite plan to “cure whatever ills mankind endure”:

Death, (grinning,) thus address’d the three:—
“‘Birds in their little nests agree,’
But you are all so jealous grown,
Like dogs, when snarling for a bone,
That one would think all sickness fled,
And I had hid my ghastly head.
I truly feel to each of you,
From me deep gratitude is due.”