Some of the natives of the Canary Islands did fish in the water surrounding the islands. A lot of the English scholarship on the natives of the Canaries is outdated or simply unhelpful (Cruz, "Towards and English Bibliographical List of the Canary Islands", Atlantis (December, 2000), pp. 221-239). However, a significant number of older histories and travelers accounts are translated and available online.

John Abercromby writes in his (very early) analysis of pottery on the Canaries that the inhabitants of Ferro (El Hierro) did fish, and is the only island to have "shell-mounds.," ("The Prehistoric Pottery of the Canary Islands and its Makers", The Journal of the Royal Anthropological Institute, 1914, p. 318).

The older histories of the Canary Islands also make reference to their fishing. Two French clergymen wrote a history of the conquest of the Canaries by Jean de Bethencourt in beginning in 1402 sometime in the mid-fifteenth century. The two wrote that the natives of the Canaries had two methods of fishing - "one spearing by torchlight; the other by poisoning the water with the juice of the tabaiba," a type of local flowering plant. George Glas, an English mariner, translated and published a seventeenth century Spanish account of the Islands, that also mentions the natives fishing - "They were excellent swimmers; and used to kill the fish on their sea-coasts with sticks" (The History of the Discovery and Conquest of the Canary Islands, 1767 ed. p. 17; ODNB Online "George Glas," by Baigent).

In the preface to the Hakluyt edition of The Canarian, which details the conquest of the Canaries, a report by Gomez Azurara in 1448 states that "the people of Palma had neither bread nor vegetables, but lived on mutton, milk, and herbs; they did not even take the trouble to catch fish like the other islanders," (p. xxxi).

If you're curious about the Canaries, I would say the two main places to check are Mercer's The Canary Islanders: Their Prehistory, Conquest and Survival and Fernández-Armesto, The Canary Islands After the Conquest.