The term 'Christian' refers to those who follow the teachings of Jesus, most importantly his Messianic fulfillment.
Thus, considering that the term is anachronous to Jesus' birth, we can theorize that, no, Jesus was not a 'Christian at birth.
But, more fundamentally speaking, considering that Jesus had to achieve perfection, first through studying the Scriptures, and then acting upon his convictions, he wouldn't have even been aware of his pre-eminence as a new born, or within the first few years of his life.
Jesus acquired knowledge, wisdom and understanding throughout his life, he was not endowed with omniscience, nor was he God, ever.
So, no, only when Jesus became spiritually mature did he realize his purpose on earth and his future destiny.