The phrase green thumb denotes considerable talent or ability to grow plants.—Synonym: green fingers.

Here, the adjective green refers to the colour of growing vegetation.
—Cf. also to be not so green as one is cabbage-looking.

These are, in chronological order, the earliest occurrences of the phrase green thumb that I have found—incidentally, this shows that, contrary to what is habitually asserted, for example by the Oxford English Dictionary (online edition, June 2021), this phrase is not of American-English origin:

1-: From My Lady in Town. A Letter of Feminine Fashion and Thought. By Miss O’Conor Eccles, published in the Leicester Chronicle and The Leicestershire Mercury (Leicester, Leicestershire, England) Saturday 20^th July 1907:

Garden Pests.
A friend of mine who loves—I had almost written lives for—her garden, tells me of a drastic remedy for the green fly, which she has lately applied with good results to a very young, and very much treasured rose tree which had been recently bought through the post, and which she feared would contaminate its neighbours. Spraying had failed, drenching had failed, it was a case of kill or cure, so she pulled the whole plant up bodily early in the morning before the sun was strong, and laid it under water in the bath—still slightly warm and very soapy—from which she had herself just emerged, and kept it under water for about five minutes. She then put it back in the garden, keeping it covered with a barrel to keep off the sun for three days. At the end of the time there was not a green fly to be seen, the rose was blossoming unchecked by its treatment, and its neighbours were saved from possible contagion. Of course, one could not apply such treatment to a whole garden, nor would one recommend it to the average gardener, but my friend has what old country women call “a Green Thumb,” that is to say, the gift of making anything and everything grow. Plants will endure liberties at her hands which a less affectionate, though more scientific gardener could not attempt.

2 & 3-: From articles on gardening, by ‘B. P. S.’, published in The Hampshire Advertiser (Southampton, Hampshire, England):

2-: Of Saturday 16^th July 1921, under the title Flowers in Hampshire. Struggle with the Drought—this article also contains the earliest occurrence of the adjective green-thumbed that I have found:

“My garden is an ash-heap,” said a very charming elderly lady to me the other day. Even she, who was renowned for her “green thumb,” had to give in to some mightier thumb than her’s [sic] pressed upon the water-taps of heaven to prevent one drop from falling.
[…]
With the green-thumbed lady, we may say of the “flower-pot” strip—it is an ash-heap.

3-: Of Saturday 4^th February 1922, under the title Violets in Hampshire. The January Bloom:

Someone came into the room just now where I was sitting, and deliberately sniffed, and sniffed again. I was not perturbed, did not think it was the first symptoms of “flu,” and rush for the eucalyptus bottle. I knew a better explanation than that, and sniffed in company.
Mid-January—the dormant time for all flowers, even frame violets, so the books tell you—yet on the mantel-piece stood a vase with at least six large full-blown blossoms of the Princess of Wales. They swayed in all sorts of graceful positions on the longest of strong stalks, mingling with a few healthy, fresh green leaves. A cheerful wood and coal fire was warming the room to perfection. Outside, the black dripping sky spread its gloomy pall, but inside how those dainty purple blooms scented the room with an ease that was little short of marvellous.
No wonder the visitor sniffed, then went closer and examined, then fell to sighing, and finally said, ”How do you do it?”
I told her this was Hampshire; I could not take all the credit, not nearly all. She said she knew it was Hampshire. I said it was the southernmost portion of South Hampshire. She said she knew that, too, having just to her cost, her very great cost, moved all her furniture, herself and her personal belongings from one end of England to the other. If the “other,” however, was going to grow violets like those, pointing to the mantel-piece, it would re-pay her twice over. But violets like those, in January, she must have.
I told her it was not difficult. One had to do a little of course, but Nature, warm-hearted South Hampshire Nature, did most of it. “To begin with,” I said, “these particular ones were sheltered in a cold frame, but I have picked violets at this time of year, only a little smaller, out in the open.”
“I want to know how you do it, right away from the beginning,” she said.
“There are books,” I began—“violet-culture, you know—it’s rather a big subject.”
“I thought you said it wasn’t difficult,” she murmured.
“No, more it is,” I said, “if you have a green thumb, and love them, and have a frame, and no gardener, and a little leaf-mould, and a crumb or two of chicken manure, and live in South Hampshire.”
“I think I can supply most of that,” said my visitor, but I heard her muttering “green thumb” to herself in a doubtful manner, and I feared—I feared just a very little for her violets.

4-: From The Yorkshire Evening Post (Leeds, Yorkshire, England) of Thursday 27^th November 1924:

“THE GREEN THUMB.”

The guest at the Leeds Women’s Luncheon Club to-day was Mrs. Marian [sic] Cran *, author of popular gardening books, and well known to amateur gardeners who have wireless sets for her interesting gardening talks.
She gave an address on “The Green Thumb”—the facility which the plant-lover has for making the seed, the cutting, and the plant grow successfully—and advised all gardeners to cultivate the green thumb by patience, self-control, understanding, and love of the earth.

* Marion Cran (née Marion Dudley – 1875-1942) was a popular gardening writer and the first woman gardening broadcaster.

5-: From This Morning’s Gossip, by ‘The Onlooker’, published in the Leeds Mercury (Leeds, Yorkshire, England) of Friday 28^th November 1924:

The Green Thumb.
Mrs. Marian [sic] Cran’s address to the Leeds Women’s Luncheon Club yesterday was a very delightful one.
Its title, “The Green Thumb,” had been puzzling members for a week, but Mrs. Cran solved the mystery right away by explaining that a green thumb, or a growing finger, was ascribed to lovers of plants and flowers whose garden work was always a success no matter whether they did things by rote or not.

6-: From To-day’s Gossip, by ‘The Rambler’, published in the Daily Mirror (London, England) of Saturday 6^th December 1924:

The Green Thumb
If you don’t know what the “Green Thumb” is you will be able to learn to-morrow night, since Mrs. Marion Cran is talking about it at the Bermondsey Bookshop. The owner of this is Leon M. Lion’s sister, Mrs. Gutman, who arranges Sunday evening lectures here and finds that the popular audiences sit rapt under some very fine speakers. Mrs. Cran is an expert on gardens, and Bermondsey is great on gardens, too, of the window-box order, and has a gardening branch under the London Gardens Guild.

7-: From Opus 7 (London: Chatto & Windus – New York: The Viking Press – 1931), by the English novelist, short-story writer and poet Sylvia Townsend Warner (1893-1978)—Opus 7 is a narrative poem, set in the English countryside, about Rebecca Random, a drunken woman with considerable skill in making plants grow:

Some skill she had, and, more than skill, a touch
that prospered all she set, as though there were
a chemical affinity ’twixt her
stuff and the stuff of plants—a pulse, a stroke,
implicit in her grubbing paws, that woke
the dreaming seed, and bade the root take hold;
an Open Sesame of the hand, that old
west-country parlance knows as “a green thumb.”

Perhaps Opus 7 introduced the British-English phrase green thumb into American English. For example, the following is from an article about the Hemet Woman’s Club Spring Flower Show, published in The Hemet News (Hemet, California, USA) of Friday 23^rd March 1934:

Sylvia Townsend Warner, in a poem in blank verse, tells of an English country legend that those who are successful with flowers possess a “Green Thumb.”
“Some skill she had, and, more than skill, a touch that prospered all she set, as though there were a chemical affinity twixt her stuff and the stuff of plants—a pulse, a stroke implicit in her grubbing paws that woke the dreaming seed and bade the root take hold; an Open Sesame of the hand, that old west-country parlance knows as ‘a green thumb’.”
Those with the “green thumbs,” or those who desire to be “green thumbs,” should enter their yards, gardens or flowers in the Woman’s club Spring Flower Show in April.