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

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

The earliest occurrences of the phrase green fingers that I have found are:

1-: From “Oh, Come ye in Peace”, in The Misses Make-believe (New York: Henry Holt and Company, 1906), by the Scottish author Mary Stuart Boyd (née Mary Kirkwood – 1860-1937):

Though the chill dark days of winter had proved dreary enough to the town-dwellers, with the advent of spring an unwonted excitement awoke in the exiles’ hearts. To them everything was new, even wonderful. Their previous experience of the country had centred in autumnal visits to seaside resorts, where the nearest approach to Nature lay in the gardener-tended flower-beds bordering the esplanades. Now the wonder-book of the spring was opening its countless leaves before them.
For the first time in her life Eileen had been able to gratify her latent gift for gardening. Her lack of experience was more than counterbalanced by the dual possessions of boundless enthusiasm, and of what old wives call “green fingers”: those magic digits that appear to ensure the growth of everything they plant.

2-: From The Daily Mirror (London, England) of Wednesday 18^th April 1923:

THE FLAT AND ITS FLOWERS.
HOW THE LONDONER GOES A-GARDENING.

Growing something! All the civilisation in the world doesn’t get rid of the desire in the average man, woman and child to create, somehow, a garden.
And a tub or two on the balcony of a flat, a window-box or a shelf of growing plants, often affords as much pleasure as a whole garden—and as much anxiety, too! If you have what country people call “green fingers,” you will always succeed.

3-: From The Halifax Daily Courier and Guardian (Halifax, Yorkshire, England) of Thursday 8^th March 1928:

Mrs. Marion Cran *, the well known writer on gardens and gardening, [was] speaking at the meeting of Halifax Women’s Luncheon Club yesterday on the growing of plants in smoky districts […].
[…]
In passing, Mrs. Cran referred to people who had “green fingers.” This, she explained, denoted that they possessed a peculiar sympathy with plants and flowers, a real mental tenderness for plants which could not help expressing itself.

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

The adjective green-fingered means having considerable talent or ability to grow plants. The earliest occurrences that I have found are:

1 & 2-: From the column Listen, Girls!, by Jessie Roberts, published in the Los Angeles Express (Los Angeles, California, USA)

1-: Of Monday 5^th January 1914—Jessie Roberts writes the following after quoting a letter by ‘L. N.’:

L. N.’s statement that she has a genius for making things grow reminds me of the charming English expression for that particular gift. Such people are called “green fingered.” And it is wonderful how people vary with flowers. How those green fingered ones can induce blossoms to appear in the most desert places; how their window boxes will be a riot of bloom while another person, apparently taking just the same care of and feeling quite as much love for flowers, will not be able to induce even a geranium to respond with a single scarlet flower.

2-: Of Monday 2^nd March 1914:

J. L. C., who sends an address, adding that she has had fine plants from the house given, goes on to say: “But, alas! I am not green fingered . . . who can tell me of a book of instructions?” Perhaps some of the readers will know a good book on violet growing. If they do, and send me the title, I will see that J. L. C. gets it.

3-: From the following advertisement, published in The Philadelphia Inquirer (Philadelphia, Pennsylvania, USA) of Saturday 26^th May 1917:

In older countries you may hear of “the green-fingered folk”—meaning people who have only to touch growing plants to make them thrive. The Garden Store, with its garden seeds and tools, is a Paradise for such folk.

4-: From Colour in My Garden (New York: Doubleday, Page & Company, 1918), by the U.S. author Louise Beebe Wilder (1878-1938):

Under the care of our green-fingered grandmothers gardens throve and were full of hearty, wholesome colour and searching fragrance; and they breathed subtly the gentle personality of those whose rare leisure was spent in digging and pruning, weeding and dreaming among the plants they loved so well.