There is an entry in Ancestry.com - Records for David Adams born Coleraine, Northern Ireland in 1800. It names his parents as John Adams and Ann. David married Ann Roberts (1818- ) and had five children- John, William, Alexander, Thomas and Ann.
There may be more on Ancestry.com, which I cannot access as I am not a paying subscriber.
On his website, which focuses mainly on William Adams, his 16 children and their descendants, Graeme Adams includes a note about the forebears of David. You can pick up that website by googling KiwiAdams.