Trying to Find Information About My Great Great Grandmother Mary Verdon

Am looking for Mary Verdon the mother of my great great grandfather on his death certificate her name is shown as Mary Gallagher so some where between1832 and 1916 she became a Gallagher I have no name so don't know where to start looking I can not find a record of the birth of James Verdon who said he was born at Parramatta nsw the first record, is his marriage to Margaret Sheehan at Conargo nsw in 1861.
He called himself Verdon so I guess that his mother was Verdon and later married a Gallagher but my great great grandfather did not take his name and didn't except it to show on any official records this means that I have no male name to look for and a span of 20 odd years