Muslims believe in Jesus PBUH's Virgin birth because Quran testifies this fact. In fact in Quran there is an entire Chapter dedicated to Mary (Chapter 19). I will quote few verses from it:
And mention, [O Muhammad], in the Book [the story of] Mary, when she withdrew from her family to a place toward the east.
And she took, in seclusion from them, a screen. Then We sent to her Our Angel, and he represented himself to her as a well-proportioned man.
She said, "Indeed, I seek refuge in the Most Merciful from you, [so leave me], if you should be fearing of Allah ."
He said, "I am only the messenger of your Lord to give you [news of] a pure boy."
She said, "How can I have a boy while no man has touched me and I have not been unchaste?"
He said, "Thus [it will be]; your Lord says, 'It is easy for Me, and We will make him a sign to the people and a mercy from Us. And it is a matter [already] decreed.' "
So she conceived him, and she withdrew with him to a remote place.
And the pains of childbirth drove her to the trunk of a palm tree. She said, "Oh, I wish I had died before this and was in oblivion, forgotten."
But he called her from below her, "Do not grieve; your Lord has provided beneath you a stream.
And shake toward you the trunk of the palm tree; it will drop upon you ripe, fresh dates.
So eat and drink and be contented. And if you see from among humanity anyone, say, 'Indeed, I have vowed to the Most Merciful abstention, so I will not speak today to [any] man.' "
Then she brought him to her people, carrying him. They said, "O Mary, you have certainly done a thing unprecedented.
O sister of Aaron, your father was not a man of evil, nor was your mother unchaste."
So she pointed to him. They said, "How can we speak to one who is in the cradle a child?"
[Jesus] said, "Indeed, I am the servant of Allah . He has given me the Scripture and made me a prophet.
And He has made me blessed wherever I am and has enjoined upon me prayer and zakah as long as I remain alive.
And [made me] dutiful to my mother, and He has not made me a wretched tyrant.
And peace is on me the day I was born and the day I will die and the day I am raised alive."
That is Jesus, the son of Mary - the word of truth about which they are in dispute.[Quran 19:16-34]
Also note there is no description in entire Bible how Mary defended having a baby without any interaction with a man. In Jewish law, she should have been stoned to death. But still there is no explanation in Bible how Mary got away with having a Virgin birth.
Quran on the other hand, mentioned that Allah gave power to Jesus PBUH to speak when he was still a newborn and that's how he defended his mother. This story of Jesus speaking when he was still a baby is not in Bible either.
Quran does not leave any space for doubts. It states clearly that Jesus PBUH is son of Mary. This is to deny that Jesus is not son of God. Quran also does not use term son for his true believers as Bible does because it leads us to where Christianity is today. Bible has called so many prophets son of god and then they discovered a separate term for Jesus as begotten son even though in Hebrew the word for son and begotten son are same. Quran does not call god the father either for the same reasons.
Regarding Joseph husband of Mary per Biblical accounts, there is no mention of him in Quran or hadith. So I can't say if he was actually married to Mary or if this is something invented by Christians to defend their false ideology.
Sorry if this hurt feelings of any Christian brothers and sisters. I am just stating facts from my POV.