Hello KDQ,
His wife's name was Adriana Heyliger - she was Dutch, and he married her on St Eustatius.
Since posting this, I've discovered something on Rootsweb which suggests that he may have been born on St Eustatius. So that it may be his father who went there from Ireland, maybe in the first half of the 18th century. The family appear to have owned a sugar plantation there. I know nothing about his father, except that I think that he was also William Moore!
I have something from a family member which suggests that these Moores may have been a related branch of the family who were Earls of Drogheda - not in a direct line of descent, but cousins. I can't confirm this.
It is so hard to do research either in Ireland or the West Indies from Australia - this is why I'm hoping for a lead or two!
Thanks,
MarieC