Is there a specific prayer for requesting a child, such as a two-rak'ah prayer in which the Ibrahimi prayer is recited in prostration, followed by supplication for a child?

There is no evidence from the Sunnah or the actions of the righteous predecessors for a specific prescribed method of supplicating for a child during prayer. The best guidance is the guidance of Muhammad, peace and blessings be upon him. Whoever desires a child should supplicate, for Abraham and Zechariah, peace be upon them, resorted to their Lord with supplication, and Allah granted them offspring. Prostration is one of the greatest occasions for supplications to be answered, based on the Prophet's saying, peace and blessings be upon him: "The closest a servant is to his Lord is while he is prostrating, so increase in supplication." The majority of scholars consider the prayer of need (Salat al-Hajah) to be permissible for requesting a child or anything else.