There are no prescribed prayers in the sources for this, but you can make use of any general prayers from the Qur’an and the Sunnah for this purpose. You are recommended to earnestly persist in asking Allah to grant you offspring. Make Du`aa’ in the good times such as in prostration, in night vigil, etc. At the same time, you should perform Prayer and obey Allah, Most High. Obedience of Allah and reciting the Qur’an and Du`aa’ are means of attaining Allah’s favors.

There are two supplications which we can infer from the Qur’an. We quote them below:
Rabbi laa tadharnee fardan wa anta khayru al-waritheen
(My Lord do not leave me childless; You are the best of heirs).
Rabbi hab lee min ladunka dhurriyyatan tayyibatan innaka samee’ al-ddu’ai
(My Lord, grant me a righteous offsrping as a pure gift of Yours; for verily You are the Hearer of prayers).
Both of the above prayers are from Prophet Zakari’ah who prayed for a progeny; Allah heard his prayers and granted him his son Yahya.

Also, we are advised to keep on making istighfar as prophet Nuh advised his people to as the Qur’an narrates:
“And I have said: Seek pardon of your Lord Lo! He was ever Forgiving. He will let loose the sky for you in plenteous rain, and will help you with wealth and sons, and will assign unto you Gardens and will assign unto you rivers.” (Nuh: 10-12)

So believe in the promise of Allah and pray sincerely, seeks and also follow up your supplications with penitence, and lots of charities, for charities are truly effective and beneficial in helping us attain and realize our aspirations and prayers.

Finally, you should not ignore medical check along with du`a’ to have children. The Prophet, peace and blessing be upon him, instructed us: “Seek medical treatment, for Allah has provided a cure for every illness.” This Hadith should be taken as encouragement to seek medical treatment along with making Du`a’.