Are there verses of the Quran you read for the purpose of easing child birth for a woman?
I do not know of anything about that from the Sunnah. However, if a person reads upon a pregnant woman who is in labor, verses which indicate easiness like:
يريد الله بكم اليسر ولا يريد بكم العسر
Allah intends for you ease, and He does not want to make things difficult for you
And verses that speak about pregnancy and childbirth like His statement:
وَمَا تَحْمِلُ مِنْ أُنْثَى وَلا تَضَعُ إِلَّا بِعِلْمِه
And no female conceives or gives birth, but with His Knowledge.
And His statement:
إذا زلزلت الأرض زلزالها وأخرجت الأرض أثقالها
When the earth is shaken with its (final) earthquake.
Verily this is beneficial and helpful by the permission of Allah. The Quran is all a cure. If the recitor and the one being read to are believers in its effect, then no doubt there will be an effect because Allah says:
وننزل من القرءان ما هو شفاء ورحمة للمؤمنين ولا يزيد الظالمين إلا خسارا
And We send down from the Quran that which is a healing and a mercy to those who believe (in Islamic Monotheism and act on it), and it increases the Zalimun (polytheists and wrong-doers) nothing but loss.
This verse is general; the mercy and cure includes the cure of the hearts from the diseases of doubts and the curse of the body from serious sickness.
Faisal ibn Abdul Qaadir ibn Hassan