The short answer
No. The Galapagos gave him a box of anomalies he could not throw away, not an idea. The idea came in London, in 1837 and 1838, after other people had sorted what he brought home.
What happened on the islands
The vice-governor of the colony, an Englishman named Lawson, told Darwin that the tortoises differed from island to island, and that he could tell which island any one came from. Darwin, in his own words, did not for some time pay sufficient attention to this statement, and had already partly mixed the collections from two islands.
The finches
He brought the small birds home as a mixed bag, taking some for wrens, some for finches, some for blackbirds. In London in 1837 the ornithologist John Gould told him they were a closely related group of ground finches found nowhere else, and that the mockingbirds from different islands were separate species.
The famous sentence came later still, in the 1845 edition of his travel journal: one might really fancy that one species had been taken and modified for different ends.
What actually convinced him
His own pocket book for 1837 says what struck him: the character of the South American fossils, and the species of the Galapagos. Within weeks of opening his first notebook he listed the proofs that animals on separate islands become different: the Galapagos tortoises, the mockingbirds, the foxes of the Falklands and Chiloe, and the hares of England and Ireland. Not the finches.