There is some conjecture about the first European birth on Australian soil, but William Nash seems to be the prime candidate.
His father, also William, was a Marine Private and his mother was Maria Haynes, who came as his common-law wife (they were not married until later). There are biographical entries on both parents in The Founders of Australia: A Biographical Dictionary of the First Fleet, by Mollie Gillen, published in 1989 (pp. 261-262). The book also lists those who were born on the voyage.
William Nash was baptised on 25 May 1788 (his actual birth date is not known), and died on 19 June 1789. Digitised copies of his baptismal and death register entries can be purchased from the NSW Registry of Births, Deaths and Marriages.
Governor Phillip's reports are published in The Historical Records of Australia Series 1 Volume 1, but there is no mention of the first birth in the colony; on 12 February 1790 he simply states the number of births (59) and deaths (72) that had occurred to date.
Unless further records are discovered, we will probably now never know definitely who was the "first born", but William Nash is the most likely.