There was a big controversy over this in the fifth century. Nestorius, the Patriarch of Constantinople, objected to the popular phrase “Mother of God” and said that “Mother of Christ” would be a better way to refer to Mary. He was removed from his position and exiled to a monastery in the Egyptian desert.
Well, the phrase “Mother of God” had already been in use for pagan goddesses. Elizabeth, in Luke 1:43, refers to Mary as “the mother of my Lord”, but not by the pagan term.