Did Jesus speak when he was a baby?

No. The Bible makes no reference to this or of Jesus being in any way different to a normal baby except for the manner of His conception.

Answer:

Only two ancient religious texts speak of Jesus at all. The Bible and The Qur'an.

The Qur'an foretells that Jesus will speak as a baby (He shall preach to men in his cradle and in the prime of manhood, and shall lead a righteous life. Sura 3:46) . It later states that Jesus spoke as a baby stating "I am indeed a Servant of Allah. He will give me revelations and make me a Prophet." 19:29-30).

The Apocryphal books of the Bible support this story in the Infancy Gospels. These books were not included in the Bible as it is compiled today

Talking baby myths are not unique to these two religions. The Buddha is aid to have walked and spoken immediately after his birth (circa 600 BCE) saying, after he took s seven steps north "I am chief of the world, Eldest am I in the world, Foremost am I in the world. This is the last birth. There is now no more coming to be."

European myths of talking babies usually source such mystical event as the child being a changeling.

This question in some way supposes that Jesus was different to other babies and that he had divine powers that would allow him to do what other babies could not.

However, there is one indisputable reason that Jesus did not and could not utter intelligible words as a baby. The larynx of a baby is high in the throat, just as it is in all other mammals, allowing the baby to suckle and breathe at the same time. As we grow older, the larynx descends, creating a larger cavity above the larynx, called the pharynx. Once the larynx descends, we can no longer swallow and breathe at the same time. It is the pharynx that allows humans to make the complex sounds necessary for speech. Apart from all other reasons for a baby not being able to speak, either his larynx had not yet descended and he could not speak, or it had descended and he could not suck milk.