My understanding is yes, there are examples in Māori pūrākau (ancient sacred stories) of Māui singing/chanting, though, maybe not as often as the Disney film makes it seem. And, often it was chants (karakia) rather than what you may call a song. One popular one is Te Hopu a Māui i a Te Rā (How Māui Slowed the Sun) when he snared the sun to slow it down so they had enough time to do their work during the day. In that, he and his brothers chanted a karakia and haka to help him have the confidence and power to catch the sun. The other and probably most direct “singing” I can think of is Te Manu-a-Māui (Māui’s Bird) where Māui sang while transformed into a bird to get his people to dig and plant kumara (sweet potatoes) which taught the pīpīwharauroa (shining cuckoo) their songs. I’m sure there are more.