The best place to look for records of these folk before they emigrated is
www.scotlandspeople.gov.uk.
According to the marriage record, both James Miller and Julia Maria Margaret McMillan were born in Toronto. His parents were John Miller and Isabel, and Julia's were Robert McMillan and Anna.
I see in the 1881 census in Toronto a family consisting of Robert McMillan, 48; Annie McMillan, 43; and John, 22; Robert, 18; Charles, 16; Thomas, 11; James, 9; Agnes, 7; and Alexander, 4.
There are index listings at FamilySearch of the deaths of two sons of Robert McMillan and Ann Roy; Charles, born 14 February 1865, d 1936, and Alexander, born 30 January 1877, d 1956, who would match the ages in the 1881 census. So I think it is likely that Julia's mother's name is Ann or Anna Roy.
Robert McMillan and Ann Roy were married in the parish of Anstruther Wester, county of Fife, Scotland, in 1857. You can view their marriage certificate online at Scotland's People for a modest fee. It will tell you the names of their parents including their mothers' maiden surnames.