This visit to her aunt house is not the story of her birth nor the assigning of Zachariah to be her guardian.
Prophet Zacharia as the guardian of Mary:
When Ann, whose husband was Imran the father of Mary (mother of the Christ), became pregnant, she vowed that what was in her womb would be dedicated to serve in the House of God at Jerusalem.
This is in the Quran 3: 35 - 37
إِذْ قَالَتِ امْرَأَةُ عِمْرَانَ رَبِّ إِنِّي نَذَرْتُ لَكَ مَا فِي بَطْنِي مُحَرَّرًا فَتَقَبَّلْ مِنِّي إِنَّكَ أَنتَ السَّمِيعُ الْعَلِيمُ . فَلَمَّا وَضَعَتْهَا قَالَتْ رَبِّ إِنِّي وَضَعْتُهَا أُنثَى وَاللّهُ أَعْلَمُ بِمَا وَضَعَتْ وَلَيْسَ الذَّكَرُ كَالأُنثَى وَإِنِّي سَمَّيْتُهَا مَرْيَمَ وِإِنِّي أُعِيذُهَا بِكَ وَذُرِّيَّتَهَا مِنَ الشَّيْطَانِ الرَّجِيمِ .
فَتَقَبَّلَهَا رَبُّهَا بِقَبُولٍ حَسَنٍ وَأَنبَتَهَا نَبَاتًا حَسَنًا وَكَفَّلَهَا زَكَرِيَّا كُلَّمَا دَخَلَ عَلَيْهَا زَكَرِيَّا الْمِحْرَابَ وَجَدَ عِندَهَا رِزْقاً قَالَ يَا مَرْيَمُ أَنَّى لَكِ هَذَا قَالَتْ هُوَ مِنْ عِندِ اللّهِ إنَّ اللّهَ يَرْزُقُ مَن يَشَاء بِغَيْرِ حِسَابٍ
The interpretation by the late Mohammed-Ali Hassan:
>> "([Remember] when the wife of Imran) her name was Ann daughter of Faqoutha; her husband called Imaran, son of Mathan, the father of Mary,
>> (said, "Lord! I have vowed to You what is within my womb [that I am pregnant with], to be dedicated to You [: consecrated exclusively to Your service, and I should not use him for my advantage], so accept [this vow] from me. Surely, You are the All-Hearing [of what I say], the All-Knowing [of my intention].)
>> (When she was delivered, [she saw it was a female]
>> ([So Ann] said: "My Lord! I am delivered of a female" [then how can I let her serve together with the men in the temple?] )
>> (–– God knew very well what she was delivered of –– [because it is He Who created and made her]);
>> ("a male is not like a female" [in comparison];
>> ("I have named her Mary, and I commend her and her progeny to Your protection from Satan, the 'rejected and cursed'.[lest he should deceive her]")
>> (Right graciously did her Lord received the [child] with gracious favor [i.e. He accepted her vow and prepared everything for that, even though she was a female] ):
>> (He made her grow a goodly growth) [ with no disabling defect or harmful disease; then when she grew and was twelve years of age, her mother brought her to the Jewish rabbis in the House of God at Jerusalem, and said: I present the vowed daughter of mine to you; therefore they competed for who will be her guardian; Zachariah said: I am more worthy with her than all of you, because her aunt is my wife; but they refused, and said: We do not accept unless we go and make lot {in a way of consulting God}; so they went to Jordan river; they were twenty nine men, deciding that they cast their pencils in the river, then anyone whose pencil be fixed in its place, and not move with the stream of the river, then he is chosen to be her guardian; then the pencil of Zachariah was fixed in its place and did not move with the stream of water; so he took her with him, build a room for her in the temple with a ladder, and none might enter her room save him, and made a prayer chamber for her in which she might pray and worship God; he also brought for her the food, water and all her needs.]:
>> (He assigned her to Zachariah to take charge of her [so he became her guardian] ):
>> (Every time that Zachariah entered into her prayer chamber, he found her having a provision) of the fruit. Some rabbis and preachers threw some fruits for her
>> ( [Zachariah] said: "O Mary! How [comes all] this [fruit] to you?")
>> ( [Mary] said: "From God [He makes the hearts of righteous people kind to me, and they bring to me these fruits]):
>> (for God offers provision to whom He pleases, without stint.")
"Know, [reader,] that Christians do not know the story of the birth of Mary, neither do they know the name of her father, in spite of that she is the mother of their prophet, while it is detailed in the Quran; and this is one of the miracles of the Quran."