I have read the (abridged) Journals of James Cook's three voyages of exploration in the Pacific, and I think I can share some information that I hope answers your question.

To my memory (and others may feel free to correct me) Cook never recorded crossing paths with a Polynesian canoe while voyaging across open ocean, with good reason. First, the Pacific ocean is just massive, and the chances of crossing paths with a vessel of any kind are incredibly slim. Second, by the time European exploration of the Pacific began, Polynesian journeys between island groups were not as common as they had been during the initial period of settlement and expansion.

That being said, Cook was certainly aware of the Polynesian's capabilities as navigators and seafarers, and was genuinely impressed with their vessels. On his first voyage they took onboard a Polynesian navigator and priest named Tupaia from the Island of Raiatea. On Thursday, July 13th, 1769 Cook related the following in his journal:

we found him to be a very intelligent person and to know more of the Geography of the Islands situated in these seas, their produce, and the religion laws and customs of the inhabitants then any one we had met with.

Tupaia indeed had extensive knowledge of the islands beyond Raiatea proper, and served as both a translator and navigator. He was able to provide a chart of around 74 islands around Raiatea. A picture of Cook's copy of the chart can be found here, although it is important to note that some island groups are "misplaced" in that Polynesian directional terms and map making differed from European tradition. Tupaia intended to return to England with Cook, but died en route in Batavia in 1891770. Upon his death, Cook remarked that he was "a Shrewd Sensible, Ingenious Man..."

Cook encountered Polynesian canoes at every inhabited Island he visited. However, the most memorable encounters seem to be during his visit to Tahiti in 1774. On the 26th of April he recounts seeing a massive Tahitian war fleet, consisting of over 300 vessels. He counted about 160 "large, double Canoes, very well equip'd, Man'd and Arm'd". He also counted "170 Sail of Smaller double Canoes, all with a little house upon them and rigg'd with Masts and sails which the others had not." He was impressed enough to inquire after the "Admiral" of said fleet, and his journal seems to indicate that he was exceedingly interested in the logistics and operation of it. A few weeks later, as they prepared to depart Tahiti, Cook accompanied a Tahitian Chief named Tu to see a large double-canoe that was being constructed and nearly ready for launch. "She was by far the largest I had seen at any of the isles," Cook remarked. He asked Tu that the Canoe "might be call'd Britanne (Brit-ania), the name they have adopted for our Country." Tu agreed. Cook also provided the Canoe with an English Jack and Pendant and instructed Tu in how to properly use it.

Between the information he gathered from Tupaia, as well as his extensive interaction with Polynesian people and their canoes, it seems safe to assume that James Cook was certainly aware of the native Polynesian's seafaring abilities, even if he never encountered a canoe on the open ocean.

All quotes taken from James Cook: The Journals, Penguin Classics, 2003.