Abdullah bin al-Zubayr (may Allah be pleased with him) was the first child born in Islam in Madinah

It is narrated from Sayyiduna Ibn Abbas (may Allah be pleased with them both); he said: The Islamic calendar began from the year in which the Prophet (peace and blessings be upon him) came to Madinah, and in that same year Sayyiduna Abdullah bin al-Zubayr (may Allah be pleased with them both) was born.

Sayyiduna Abdullah bin al-Zubayr (may Allah be pleased with them both) narrates: I was named after my maternal grandfather, Sayyiduna Abu Bakr al-Siddiq (may Allah be pleased with him), and I was also given his kunyah. Abdullah bin al-Zubayr (may Allah be pleased with them both) had two kunyahs: Abu Bakr and Abu Khubayb.

Hisham bin Urwah narrates from his father that when Sayyidah Asma bint Abi Bakr (may Allah be pleased with them both) migrated to the Messenger of Allah (peace and blessings be upon him), she was pregnant with Sayyiduna Abdullah bin al-Zubayr (may Allah be pleased with them both). When he was born, she brought him to the Prophet (peace and blessings be upon him) so that he (peace and blessings be upon him) would perform his tahnik. The Messenger of Allah (peace and blessings be upon him) took him in his lap and called for a date. He chewed it, then took it out of his blessed mouth and put it in the child's mouth, and performed tahnik with it. So the very first thing to enter his stomach was the blessed saliva of the Messenger of Allah (peace and blessings be upon him). The narrator says: Then the Messenger of Allah (peace and blessings be upon him) passed his hand over him and named him Abdullah. Then, at the age of seven or eight years, he came to pledge allegiance to the Prophet (peace and blessings be upon him), his father Sayyiduna al-Zubayr (may Allah be pleased with him) having told him to do so. When the Prophet (peace and blessings be upon him) saw him coming, he smiled and accepted his pledge of allegiance. He was the first child born in Madinah after the arrival of the Messenger of Allah (peace and blessings be upon him) in Madinah. The Jews (out of envy) used to say: We have cast a spell on them; now no male child will be born to them in Madinah. So when Abdullah was born, the Companions of the Messenger of Allah (peace and blessings be upon him) raised the cry of takbir. And Sayyiduna Abdullah bin Umar bin al-Khattab (may Allah be pleased with them both), when he heard the takbirs of the people of Syria who had killed Abdullah bin al-Zubayr, said: "Those who said takbir at his birth were better than those who have said takbir at his killing."
This hadith is sahih (authentic) on the condition of the two Shaykhs (Bukhari and Muslim), but they did not record it.