Results from a Google Books search strongly suggest that the expression originated (in print) in religious publications. At least in my search results, publications of the missionary wing of the Methodist Episcopal Church account for the first five matches, spread across a period of seven years. The earliest of these sources (from January 1910) credits the saying to a Commissioner McFarland, who may be the author of the third article cited below.
From "Never Paralleled in New York," in The [New York] Christian Advocate (January 20, 1910):
Many ministers have heartily indorsed the [Laymen's Missionary] Movement and will support it with their indispensable influence. With their cooperation failure is impossible, and it is the universal post-convention report from the other cities that all local causes have been helped on by the new impulse of the Movement whose watchword is "For the Other Man." As Commissioner McFarland said at the Astor dinner, The rising tide lifts every boat!
From "Japanese Church in Wonsan, Korea," in The [Nashville, Tennessee] Missionary Voice (May 1911):
Now the building of the church in Wonsan [Korea], of course, was not entirely, or maybe even mainly, responsible for the building of the great church at home. But at least it did not hinder, and who doubts that the enlargement of heart that came to them through that unselfish thing, the spiritual swing developed in that enterprise abroad, hastened and helped on the larger thing they would do at home? "The rising tide lifts all boats," and "the light that shines farthest shines brightest at home."
From Henry B. F. McFarland, "The Man by Man Rise of a Race of Men," in Association Men (January 1915):
The rising tide lifts all the boats upon it. All parts of the colored [Young Men's Christian] Association movement have shared in the new progress. Great corporations employing negro workmen—like the Newport News Ship Building Company, which has already started work, with its four thousand negro employees—see the advantage of such Associations.
From "Woman's Home Missionary Society," in Minutes of the One Hundred and Seventeenth Session of the New York Conference of the Methodist Episcopal Church held in First Church, Peekskill, N.Y. March 22, 1916 (1916):
Wherever there is an active, interested, wide-awake Woman's Missionary Organization there will be found a broadened vision, an increasing zeal along all lines, and a permeating spirit of the Christ-love that is most helpful. "The rising tide lifts all boats."
From Hugh Burleson, "The Progress of the Kingdom," in The Spirit of the Missions (May 1916):
The editor of THE SPIRIT OF THE MISSIONS is certain that he speaks not only for himself but for every official connected with any missionary enterprise when he declares that he would not, if he could, divert a single dollar from the relief of Europe. On the contrary, he rejoices when gifts for the war sufferers increase, not only because of his passionate desire that this awful suffering may be relieved, but also because he knows that "a rising tide lifts all the boats," and that every good cause should profit by the sympathy which this crying need awakens in the hearts of men and women who too long have been concerned chiefly about their own comfort and gratification.
From "A Hint from Harvard," in Harvard Alumni Bulletin (December 11, 1919):
There must be hundreds of Harvard men who on reading that [story] will directly double their subscriptions; the tale is so darned humane! I am not a Harvard man as you know, but the Top Sergeant and his wife make me double my paltry ten dollars. The psychology of the outsiders with colleges of their own to serve is quite correct. A rising tide lifts all the boats!
And finally, the cover of American Gas Monthly (March 1920) consists of the following quotation and subtitle:
"The Rising Tide Lifts All the Boats."
When the tide of public opinion swells through recognition of service well performed, all our boats will be lifted.
Basically what we have here is a slogan so successful that in a single decade it went from being a watchword—indeed almost a catchphrase—of a vigorous Christian missionary movement to becoming an inspirational proverb for members of the American Gas Association to contemplate.
Best Answer
'The grass is greener' in idiom and proverb dictionaries
Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) echoes the observation cited in user240918's question that "the grass is greener" derives from an ancient proverb:
Ammer doesn't provide a first occurrence date in English for this saying, as she often does for other idiomatic phrases. Gregory Titelman, Random House Dictionary of Popular Proverbs and Sayings, second edition (2000) does offer a first occurrence—but it is almost astonishingly late (1959):
Martin Manser, The Facts on File Dictionary of Proverbs (2002) agrees with the previous two authorities in judging the expression as "relatively recent" in its current form but ancient in its application:
Fairly strong support for this consensus view appears from the fact that G.L. Apperson, English Proverbs and Proverbial Sayings: A Historical Dictionary (1929), reprinted as The Wordsworth Dictionary of Proverbs (1993) contains no notice of the proverb. An updated version of Apperson's book, published by Wordsworth in 2006, however, has this entry:
J.A. Simpson, The Concise Oxford Dictionary of Proverbs 1982) concurs with the other authorities cited with regard to the 1959 first occurrence of "the grass is greener," but it conveniently includes R. Taverner's 1545 translation of Erasmus's version of Ovid's proverb, as part of Erasmus' Adages:
Charles Doyle, Wolfgang Mieder & Fred Shapiro, The Dictionary of Modern Proverbs (2012) finds a number of much earlier instances of the proverb:
'The grass is greener' in the wild
An Elephind newspaper database search turns up matches for versions of the same "the grass is greener elsewhere" sentiment from even earlier than 1913. From "De Lamar's Opinion: Absolutely Foolish to Start for the Klondyke Now," in the Salt Lake [City, Utah] Herald (August 5, 1897):
And from an untitled item in the [Des Moines, Iowa] Wallaces' Farmer (July 12, 1907):
Note that the 1907 instance describes the expression as "an old saying."
Conclusions
Everyone agrees that Ovid and (1500 years later) Erasmus voiced a sentiment very similar to "the grass always looks greener elsewhere." But aside from George Herbert's very similar observation 1640 regarding the sweetness of apples on the far side of a wall, few examples have come forward that demonstrate continuity between Ovid and Erasmus and the twentieth-century English proverb about greener grass. It is therefore hard to say whether the earliest occurrences of "the grass is always greener" in English represent a rediscovery/translation of old Latin wisdom or a homely independent coinage addressing a seemingly universal human trait.
In any event, it seems clear that at least some people were speaking (and writing) figuratively about how much greener the grass looks at some distance from where one stands more than a century ago—including instances from 1897 and 1907.