The conclusion that Mary did not feel pains at the birth of Christ can be deduced from her immaculate conception. Since pains at childbirth is one of the punishments inherited from original sin which Mary is free from. There are a number of passages and quotes from Church Fathers that support this conclusion, some of them are given below:
Scriptural Passages on Mary’s Painless Delivery of Christ
“Therefore the Lord himself shall give you a sign: Behold, a virgin shall conceive and bear a son, and shall call his name Emmanuel. (Isa 7:14).
This prophecy says that a virgin will not only conceive as an intact virgin but also bear her Son as an intact virgin.
Before she travailed, she brought forth; before her pain came, she was delivered of a man child (Isa 66:7)
Then he brought me back the way of the gate of the outward sanctuary which looketh toward the east; and it was shut. Then said the LORD unto me; This gate shall be shut, it shall not be opened, and no man shall enter in by it; because the LORD, the God of Israel, hath entered in by it, therefore it shall be shut (Ezk 44:1-3).
A garden inclosed is my sister, my spouse; a spring shut up, a fountain sealed (Canticles 4:12)
The interpretation by Doctors of the Church mean that the womb of the Virgin Mary is shut and sealed by God, not to be opened in natural childbirth.
The Fathers, Doctors and Popes on Mary’s Painless Delivery of Christ
St Gregory of Nyssa
Though coming in the form of man, yet not in every thing is He subject to the laws of man’s nature; for while His being born of a woman tells of human nature; virginity becoming capable of childbirth betokens something above man. Of Him then His mother’s burden was light, the birth immaculate, the delivery without pain, the nativity without defilement, neither beginning from wanton desire, nor brought to pass with sorrow. For as she who by her guilt engrafted death into our nature, was condemned to bring forth in trouble, it was meet that she who brought life into the world should accomplish her delivery with joy. (St Gregory of Nyssa, Homily on the Nativity, AD ca. 388)
St. Ambrose of Milan
Who is this gate (Ezekiel 44:1-4, see above), if not Mary? Is it not closed because she is a virgin? Mary is the gate through which Christ entered this world, when He was brought forth in the virginal birth and the manner of His birth did not break the seals of virginity. There is a gate of the womb, although it is not always closed; indeed only one was able to remain closed, that through which the One born of the Virgin came forth without the loss of genital intactness (St. Ambrose of Milan, The Consecration of a Virgin and the Perpetual Virginity of Mary, 8:52; ca. AD 391).
St Augustine of Hippo
It is not right that He who came to heal corruption should by His advent violate integrity (St. Augustine, Sermon 189:2).
In conceiving thou wast all pure, in giving birth thou wast without pain (St. Augustine, Sermon on Nativity).
Christ transcends, indeed, the miracles of all besides, in being born of a virgin, and in possessing alone the power, both in His conception and birth, to preserve inviolate the integrity of His mother: but that was done neither before their eyes nor in them. For the knowledge of the truth of such a miracle was reached by the apostles, not through any onlooking that they had in common with others, but in the course of their separate discipleship. (St. Augustine, Tractate 91:3).
Pope St Leo the Great
She brought Him forth without the loss of virginity even as she conceived Him without its loss. (Pope St. Leo the Great, Tome to Flavian, ca. AD 430)
Pope Pelagius I
Jesus Christ, true God and the same true man proceeded, that is, was born, while his mother’s virginity remained intact: for the Virgin remained such in bearing him just as she had in conceiving him (Pope Pelagius I, Letter to King Childebert I)
St. Proclus of Constantinople
O mystery! I see miracles, and I proclaim the Godhead: I perceive sufferings, and I do not deny the humanity. For Emmanuel opened the doors of nature as man, but as God did not break through the bars of virginity (St. Proclus of Constantinople, Oratio 1, no. 10; PG 65:692A).
Lateran Council (649 AD) with Pope St. Martin I
If anyone does not in accord with the Holy Fathers acknowledge the holy and ever virgin and immaculate Mary was really and truly the Mother of God, inasmuch as she, in the fullness of time, and without seed, conceived by the Holy Spirit, God in the Word Himself, who before all time was born of God the Father, and without loss of integrity brought Him forth, and after His birth preserved her virginity inviolate, let him be condemned.
Council of Toledo XVI (693 AD)
And as the Virgin acquired the modesty of virginity before conception, so also she experienced no loss of her integrity; for she conceived a virgin, gave birth a virgin, and after birth retained the uninterrupted modesty of an intact virgin.
St. John Damascene
How can death claim as its prey this truly blessed one, who listened to God’s word in humility, and was filled with the Spirit, conceiving the Father’s gift through the archangel, bearing without concupiscence or the co-operation of man the Person of the Divine Word, who fills all things, bringing Him forth without the pains of childbirth, being wholly united to God? It was fitting that the body of her, who preserved her virginity intact in childbirth, should be kept from corruption even after death. She who nursed her Creator as an infant at her breast, had a right to be in the divine tabernacles. It was fitting that she who saw her Son die on the cross, and received in her heart the sword of pain which she had not felt in childbirth, should gaze upon Him seated next to the Father. (St. John Damascene, Second Homily on the Dormition of the Mother of God)
So far as He was born of woman, His birth was in accordance with the laws of parturition, while so far as He had no father, His birth was above the nature of generation: and in that it was at the usual time (for He was born on the completion of the ninth month when the tenth was just beginning), His birth was in accordance with the laws of parturition, while in that it was painless it was above the laws of generation. For, as pleasure did not precede it, pain did not follow it, according to the prophet who says, Before she travailed, she brought forth, and again, before her pain came she was delivered of a man-child (Isaiah 66:7). The Son of God incarnate, therefore, was born of her, not a divinely-inspired man but God incarnate. But just as He who was conceived kept her who conceived still virgin, in like manner also He who was born preserved her virginity intact, only passing through her and keeping her closed (Ezekiel 44:2). (St. John Damascene, On the Orthodox Faith, IV, 14)
St. Thomas Aquinas, Common Doctor of the Church
The pains of childbirth are caused by the infant opening the passage from the womb. Now it has been said above (28, 2, Replies to objections), that Christ came forth from the closed womb of His Mother, and, consequently, without opening the passage. Consequently there was no pain in that birth, as neither was there any corruption; on the contrary, there was much joy therein for that God-Man was born into the world, according to Isaiah 35:1-2: Like the lily, it shall bud forth and blossom, and shall rejoice with joy and praise (St Thomas Aquinas, ST III, q. 35, a. 6).
St. Bonaventure, Seraphic Doctor of the Church
O God, my God: I will glorify thee by Thy Mother. For she hath conceived thee in virginity: and without travail she hath brought Thee forth (St Bonaventure,Psalter of the BVM, 62).
Roman Catechism of the Council of Trent of 1566
To Eve it was said: In pain you shall bring forth children (Gen. 3:16). Mary was exempt from this law, for preserving her virginal integrity inviolate, she brought forth Jesus the Son of God, without experiencing, as we have already said, any sense of pain. (Roman Catechism, The Creed Article III)
If you have more passages from the Doctors or Scriptures, add them below.