Duran, I do apologize for mis-interpreting your question. In re-reading your post, I can see that you are not questioning the fact that God was born, only questioning scriptural support for the December 25th date.
I found this helpful serveral years ago when I first read it:
The scriptural reference goes like this:
The Gospel of Luke indicates that the announcement of the Archangel Gabriel to Jesus' mother, Mary occurred six months after the conception of St. John the Baptist, with the date of the conception of St. John the Baptist being deduced from the prescribed time that his father, Zechariah, was serving in the temple when he received his own vision of the Archangel Gabriel. (Read the detailed explanation on this from Dr. Taylor Marshall here.)
The date of the Annunciation was celebrated by the Church since ancient times on March 25th, placing Jesus' birth nine months later in December, near the time of the winter solstice - December 25th.