The only names we have for our hero are Fitzwilliam Darcy. If he had another given name, then JA didn't bother to tell us (which the author of Jane Eyre does do). There have been conventions for naming children, but no hard & fast rules or requirements, so Darcy would not have had to have another name.

Without further evidence, I shall continue to think that his first name is Fitzwilliam and his last name is Darcy. :-)