Searching for Ulloa + Juan Fernandes, however, I could retrieve the source. It is: A Voyage to South-America by Don George Juan and Don Antonio de Ulloa, both captains of the Spanish Navy. Vol. II. London: Davis & Reymers (1758).
Chapter 4 of Book 2 begins at page 222 and is titled: "Account of the Islands of Juan Fernandes: Voyage from those islands to Santa Maria, and from thence to the Bay of Conception."
It includes the goat-dog story as follows:
"The islands of Juan Fernandes, which, on account of their situation, belong to the kingdom of Chili, are two. [...] Here are many dogs of different species, particularly of the greyhound kind; and also a great number of goats, which it is very difficult to come at, artfully keeping themselves among those crags and precipices, where no other animal but themselves can live. The dogs owe their origin to a colony sent thither not many years ago, by the president of Chili and the vice-roy of Peru, in order totally to exterminate the goats; that any pirates, or ships of the enemy might not here be furnished with provisions. But this scheme has proved ineffectual, the dogs being incapable of pursuing them among the fastnesses where they live, these animals leaping from one rock to another with surprising agility. Thus far indeed it has answered the purpose; for ships cannot now so easily furnish themselves with provisions here, it being very difficult to kill even a single goat." Juan & Ulloa (1758, p. 222 [...], 223-224)P.S.: By the way, Alexsander Selkirk has been marooned on one of the Juan Fernandes islands for four years, in 1704. He could still catch goats for his survival. After his rescue, he became the likely inspiration for the novel Robinson Crusoe.
Below is a map of the islands. Isla Alexander Selkirk is the smaller one (formerly called Más Afuera). Ironically Alexander Selkirk has never been on this island. The larger one is now called Isla Robinsón Crusoe (formerly Más a Tierra) because Alexander Selkirk's story, who has been cast away on this island for more than 4 years, is believed to have inspired the novel by William Defoe: