The argument will be made that Jesus was both fully Man and fully God, so in that sense, God (Jesus) did have a mother. That will spawn all kinds of theological discussions, I'm sure.
Trinity aside, I am hoping that respondents keep the context narrow to the Word as Flesh and not the Father nor Spirit.
Jesus, God the Son, had an earthly mother, but, God the Father, who is a Spirit, did not. Yet, the mystery of the Godhead is difficult to comprehend. There are many instances throughout the Scriptures where we get glimpses of the Trinity (word not in Bible) Godhead is - and yet it remains a mystery to our human minds.
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.