In that phrase by A.A. Hodge, he is saying that Nestorius rejected the title, "mother of god" because in fact, according to Nestorius' belief, Mary gave birth to the human nature of Christ, i.e., the human nature of Christ originated from her own flesh albeit the egg that was fertilized was by the Holy Spirit and not any human male. Put another way, Nestorius rejected any idea that God originated from Mary and came into being when Jesus was born.
Interestingly enough many today are confused in understanding John 1:1, in particular, the identity of the "logos" (Word) who John says:
John 1:1-3 (ASV) "In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made through him; and without him was not anything made that hath been made."
believe that the "Word" was Jesus, which is impossible, for Jesus of Nazareth was BORN of Mary in Nazareth, which John writes in v. 10 "And the "Logos" (Word) became flesh
and dwelt among us..." The "Logos" (Word) in John 1:1 is GOD the Son the second person of the Trinity Who was joined with a human nature (incarnation) and was born of Mary. Not only is it illogical to say that the "Logos" (Word) was Jesus, making the incarnate Son of God eternal and divine, i.e., the Son of God always existed as the God-man having a human body, etc., but grammatically it is totally impossible. This can easily be seen by substituting the word "Jesus" for "the Word" in John 1:1 and also in John 1:10 and then it would read, "and Jesus became Jesus (flesh)"
In short, this view denies the historical Jesus who was born of Mary and who until that time didn't exist (cf. Gal 4:4).