First the easy stuff, from the article you linked, it was indeed first reported by a member of Captain Cook's expedition to Tahiti (not Hawaii):
The art of surfing, known as enalu in the Hawaiian language, was first
discovered by Joseph Banks on the HMS Endeavour during the first
voyage of James Cook, during the ship's stay in Tahiti. Surfing was a
central part of ancient Polynesian culture and predates European
contact. The chief (Ali'i) was traditionally the most skilled wave
rider in the community with the best board made from the best wood.
The ruling class had the best beaches and the best boards, and the
commoners were not allowed on the same beaches, but they could gain
prestige by their ability to ride the surf on their boards
To be specific, Mr. Bank's report comes from his journal entry of May 29, 1769:
In our return to the boat we saw the Indians amuse or excersise
themselves in a manner truly surprizing. It was in a place where the
shore was not guarded by a reef as is usualy the case, consequently a
high surf fell upon the shore... but their cheif amusement was carried on
by the stern of an old canoe, with this before them they swam out as
far as the outermost breach, then one or two would get into it and
opposing the blunt end to the breaking wave were hurried in with
incredible swiftness. Sometimes they were carried almost ashore but
generaly the wave broke over them before they were half way, in which
case the[y] divd and quickly rose on the other side with the canoe in
their hands, which was towd out again and the same method repeated.
It was also reported to have been a pre-contact activity in Samoa and Tonga.
The ancient Moche (in modern Peru) also practiced a similar form that was a bit more akin to stand-up paddleboarding. We know this from archeological remains depicting identical vessels to those used for this purpose by the natives today. You might think this must be a totally independent discovery of theirs, except that at around this time somehow their culture's staple crop, the domesticated sweet potato, found its way across Polynesia. So clearly there was some amount of cultural interchange between Peru and Polynesia during this time.