This is one of only three birthdays celebrated in the liturgical year: those of Our Lady, Our Lord, and St John the Baptist.
The useful Fisheaters website offers a translation of the (non-canonical) Gospel of the Nativity of Mary, that was translated from the Hebrew by St Jerome. The sections relevant to today's feast are as follows:
The blessed and glorious ever-virgin Mary, sprung from the royal stock and family of David, born in the city of Nazareth, was brought up at Jerusalem in the temple of the Lord. Her father was named Joachim, and her mother Anna. Her father's house was from Galilee and the city of Nazareth, but her mother's family from Bethlehem. Their life was guileless and right before the Lord, and irreproachable and pious before men. For they divided all their substance into three parts. One part they spent upon the temple and the temple servants; another they distributed to strangers and the poor; the third they reserved, for themselves and the necessities of their family. Thus, dear to God, kind to men, for about twenty years they lived in their own house, a chaste married life, without having any children. Nevertheless they vowed that, should the Lord happen to give them offspring, they would deliver it to the service of the Lord; on which account also they used to visit the temple of the Lord at each of the feasts during the year.
And it came to pass that the festival of the dedication was at hand; wherefore also Joachim went up to Jerusalem with some men of his own tribe. Now at that time Issachar was high priest there. And when he saw Joachim with his offering among his other fellow-citizens, he despised him, and spurned his gifts, asking why he, who had no offspring, presumed to stand among those who had; saying that his gifts could not by any means be acceptable to God, since He had deemed him unworthy of off-spring: for the Scripture said, Cursed is every one who has not begot a male or a female in Israel. He said, therefore, that he ought first to be freed from this curse by the begetting of children; and then, and then only, that be should come into the presence of the Lord with his offerings. And Joachim, covered with shame from this reproach that was thrown in his teeth, retired to the shepherds, who were in their pastures with their flocks; nor would he return home, test perchance he might be branded with the same reproach by those of his own tribe, who were there at the time, and had heard this from the priest.
Now, when he had been there for some time, on a certain day when he was alone, an angel of the Lord stood by him in a great light. And when he was disturbed at his appearance, the angel who had appeared to him restrained his fear, saying: Fear not, Joachim, nor be disturbed by my appearance; for I am the angel of the Lord, sent by Him to thee to tell thee that thy prayers have been heard, and that thy charitable deeds have gone up into His presence. For He hath seen thy shame, and hath heard the reproach of unfruitfulness which has been unjustly brought against thee. For God is the avenger of sin, not of nature: and, therefore, when He shuts up the womb of any one, He does so that He may miraculously open it again; so that that which is born may be acknowledged to be not of lust, but of the gift of God. For was it not the case that the first mother of your nation-Sarah-was barren up to her eightieth year? And, nevertheless, in extreme old age she brought forth Isaac, to whom the promise was renewed of the blessing of all nations. Rachel also, so favoured of the Lord, and so beloved by holy Jacob, was long barren; and yet she brought forth Joseph, who was not only the lord of Egypt, but the deliverer of many nations who were ready to perish of hunger. Who among the judges was either stronger than Samson, or more holy than Samuel? And yet the mothers of both were barren. If, therefore, the reasonableness of my words does not persuade thee, believe in fact that conceptions very late in life, and births in the case of women that have been barren, are usually attended with something wonderful. Accordingly thy wife Anna will bring forth a daughter to thee, and thou shall call her name Mary: she shall be, as you have vowed, consecrated to the Lord from her infancy, and she shall be filled with the Holy Spirit, even from her mother's womb. She shall neither eat nor drink any unclean thing, nor shall she spend her life among the crowds of the people without, but in the temple of the Lord, that it may not be possible either to say, or so much as to suspect, any evil concerning her. Therefore, when she has grown up, just as she herself shall be miraculously born of a barren woman, so in an incomparable manner she, a virgin, shall bring forth the Son of the Most High, who shall be called Jesus, and who, according to the etymology of His name, shall be the Saviour of all nations. And this shall be the sign to thee of those things which I announce: When thou shalt come to the Golden gate in Jerusalem, thou shalt there meet Anna thy wife, who, lately anxious from the delay of thy return, will then rejoice at the sight of thee. Having thus spoken, the angel departed from him.
Thereafter he appeared to Anna his wife, saying: Fear not, Anna, nor think that it is a phantom which thou seest. For I am that angel who has presented your prayers and alms before God; and now have I been sent to you to announce to you that thou shalt bring forth a daughter, who shall be called Mary, and who shall be blessed above all women. She, full of the favour of the Lord even from her birth, shall remain three years in her father's house until she be weaned. Thereafter, being delivered to the service of the Lord, she shall not depart from the temple until she reach the years of discretion. There, in fine, serving God day and night in fastings and prayers, she shall abstain from every unclean thing; she shall never know man, but alone, without example, immaculate, uncorrupted, without intercourse with man, she, a virgin, shall bring forth a son; she, His hand-maiden, shall bring forth the Lord-both in grace, and in name, and in work, the Saviour of the world. Wherefore arise, and go up to Jerusalem; and when thou shalt come to the gate which, because it is plated with gold, is called Golden, there, for a sign, thou shalt meet thy husband, for whose safety thou hast been anxious. And when these things shall have so happened, know that what I announce shall without doubt be fulfilled.
Therefore, as the angel had commanded, both of them setting out from the place where they were, went up to Jerusalem; and when they had come to the place pointed out by the angel's prophecy, there they met each other. Then, rejoicing at seeing each other, and secure in the certainty of the promised offspring, they gave the thanks due to the Lord, who exalteth the humble. And so, having worshipped the Lord, they returned home, and awaited in certainty and in gladness the divine promise. Anna therefore conceived, and brought forth a daughter; and according to the command of the angel, her parents called her name Mary."