The Claim that Almighty Allah has a son has been explicitly denied in the Glorious Quran. This claim is specially made by the Christians and they say that Jesus (Peace be upon him) is the son of Allah. A delegation had come to Prophet Muhammad (Peace be upon him) from Nagran, who were talking about their claims about Triune God, that is Allah is a part of Trinity, the Father, the Son and the Holy Spirit. They also informed about few differences in these believes among various sects of Christians.
This is the reason why Allah repeatedly affirms that Jesus (Peace be upon him) is a slave of Allah like other prophets, in several places in the Glorious Quran. He affirms that it is He who molded Jesus (Peace be upon him) in his mothers womb like every other creation of His. He also affirms that He created Jesus (Peace be upon him) without a father, just like how He created Adam (Peace be upon him) without a father and a mother.
Allah the Almighty said: Allah chose Adam, Nooh (Noah), the family of Ibrahim (Abraham) and the family of Imran above the Alameen (mankind and jinns) (of their times). Offspring, one of the other, and Allah is the All-Hearer, All-Knower.
Remember when the wife of Imran said: “O my Lord! I have vowed to You what (the child that) is in my womb to be dedicated for Your services (free from all worldly work; to serve Your Place of worship), so accept this, from me. Verily, You are the All-Hearer, the All Knowing.”
Then when she delivered her (child Mary), she said: “O my Lord! I have delivered a female child,” and Allah knew better what she delivered, – “and the male is not like the female, and I have named her Mary, and I seek refuge with You (Allah) for her and for her offspring from Satan, the outcast.”
So her Lord (Allah) accepted her with goodly acceptance. He made her grow in a good manner and put her under the care of Zechariah. Every time he entered Al-Mihrab (a praying place or a private room), he found her supplied with sustenance.
He said: “O Mary! From where have you got this?” She said: “From Allah. Verily, Allah provides sustenance to whom He wills, without limit.”
(Al Quran 3: 33-37)
Here Allah announces that He has elected Adam (Peace be upon him) and the elite of his progeny who are obedient to Allah. After this he mentions the family of Abraham (Peace be upon him) which includes the sons of Ishmael (pbuh), and the family of Imran, the father of Mary.
Muhammad Ibn Ishaaq stated that he was Imran Ibn Bashim, Ibn Amun, Ibn Misha, Ibn Hosqia, Ibn Ahriq, Ibn Mutham, Ibn Azazia, Ibn Amisa, Ibn Yamish, Ibn Ahrihu, Ibn Yazem, Ibn Yahfashat, Ibn Eisha, Ibn Iyam, Ibn Rahba am, Ibn David (Dawud).
The sister of wife of Prophet Zakariyah (Peace be upon him) had a daughter named Hannah. She was married to Imran, a leader of the Israelites. For many years, she bore no children and the both remained childless. Her wish for a child increased whenever she saw another woman with a child. Although several years had passed, But she never lost hope. She believed that Allah would one day bless her with a baby, on whom she would shower all her motherly love.
She turned to the Lord of the heavens and the earth and pleaded with Him for a child. She vowed to offer the child in the service of Allah’s house, in the temple of Jerusalem. Allah granted her request. When she learned that she was pregnant, she was the happiest woman alive, and thanked Allah for His gift. Her overjoyed husband Imran also thanked Allah for His mercy.
However, Imran passed away before even seeing his child. Hannah cried bitterly on Imran’s death saying that he did not live to see his own child for whom he waited so many years.
She gave birth to a girl, and again turned to Allah in prayer: “O my Lord! I have delivered a female child,” and Allah knew better what she delivered, – “and the male is not like the female, and I have named her Mary, and I seek refuge with You (Allah) for her and for her offspring from Satan, the outcast.”
(Al Quran 3: 36)
Now Although Hannah had promised to give her child in the services of Allah’s house, she had a big problem in reference to her promise to Allah, for females were not accepted into the temple. This worried her a lot. Her sister’s husband Zakariyah, comforted her, saying that Allah knew best what she had delivered and appreciated fully what she had offered in His service. She wrapped the baby in a shawl and handed it over to the temple elders. The question of her guardianship posed a problem for the elders since it was a girl. This was a child of their late and beloved leader, and everyone was eager to take care of her. Zakariyah said to the elders: “I am the husband of her maternal aunt and her nearest relation in the temple; therefore, I will be more mindful of her than all of you.”
As it was their custom to draw lots to solve disagreements, they followed this course. Each one was given a reed to throw into the river. They had agreed that whoever’s reed remained afloat would be granted guardianship of the girl. All the reeds sank to the bottom except Zakariyah’s. With this sign, they all surrendered to the will of Allah and made him the guardian.
To ensure that no one had access to Mary, Zakariyah built a separate room for her in the temple. As she grew up, she spent her time in devotion to Allah. Zakariyah visited her daily to see to her needs, and so it continued for many years. One day, he was surprised to find fresh fruit, which was out of season in her room. As he was the only person who could enter her room, he asked her how the fruit got there. She replied that these provisions were from Allah, as He gives to whom He wills. Zakariyah understood by this that Allah had raised Mary’s status above that of other women.
Thereafter, Zakariyah spent more time with her, teaching and guiding her. Mary grew to be a devotee of Allah, glorifying Him day and night.
Ali Ibn Abi Talib narrated that the Prophet Muhammad (pbuh) said: “The best of the world’s women is Mary (in her lifetime), and the best of the world’s women is Khadija (in her lifetime).” (Sahih Al-Bukhari)
Abu Musa Al-Ashari also narrated that the Prophet Muhammad (pbuh) said: “May among men attained perfection but among women none attained perfection except Mary the daughter of Imran, and Asiya the wife of Pharaoh, and the superiority of Aisha to other women is like the superiority of Tharid (an Arabic dish) to other meals.”
[Source: The story of Jesus ('Isa): Section 1; Stories of the Prophets by Ibn Kathir]