The Bible states clearly that Joseph's mother Rachel (Genesis 30:22-24) died at the birth of Joseph's younger brother, Benjamin:

Then they [Jacob and his household] moved on from Bethel. While they were still some distance from Ephrath, Rachel began to give birth and had great difficulty. And as she was having great difficulty in childbirth, the midwife said to her, "Don't be afraid, for you have another son." As she breathed her last—for she was dying—she named her son Ben-Oni. But his father named him Benjamin. Genesis 35:16-18

The author of the Qur'an seems not to have been aware of this, when he ‘reveals’ about Jacob's family coming to Egypt:

Then, when they entered unto Yusuf (Joseph), he betook his parents to himself and said: "Enter Egypt, if Allah wills, in security." And he raised his parents to the throne and they fell down before him prostrate. And he said: "O my father! This is the interpretation of my dream aforetime! My Lord has made it come true! He was indeed good to me, when He took me out of the prison, and brought you (all here) out of the bedouin-life, after Shaitan (Satan) had sown enmity between me and my brothers. Certainly, my Lord is the Most Courteous and Kind unto whom He will. Truly He! Only He is the All-Knowing, the All-Wise. S. 12:99-100 Al-Hilali & Khan

Two Muslim commentators acknowledge that this is a problem, and seek to do damage control in their footnotes:

... His mother Rachel had long been dead, but he had been brought up by his mother's sister Leah, whom his father had also married. Leah was now his mother. They were logded with Joseph himself. (Abdullah Yusuf Ali, The Holy Qur'an, Saudi edition, fn. 1777)

According to the Biblical account - not contradicted by the Qur'an - Joseph's mother Rachel had died while giving birth to Benjamin. We may, therefore, assume that the "mother" implied in the term "parents" was another of Jacob's wives, who had brought up Joseph and Benjamin; this would be in consonance with the ancient Arabian custom of applying the designation "mother" to a foster-mother. (Muhammad Asad, The Message of the Qur'an, p. 352, fn. 96; bold emphasis mine)

Although Arabian custom is irrelevant in this case, since Jacob was not an Arab, it is certainly possible, and even likely, that Leah acted as a mother to Joseph and Benjamin after the death of Rachel, and that Joseph called her mother.

However, both Muslim commentators overlooked that this does not save the Qur'an. The Bible does not only report the death of Rachel, but also informs us that the death of Leah took place before Jacob and his household moved to Egypt. We read:

Then Jacob left Beersheba, and Israel's sons took their father Jacob and their children and their wives in the carts that Pharaoh had sent to transport him. They also took with them their livestock and the possessions they had acquired in Canaan, and Jacob and all his offspring went to Egypt. He took with him to Egypt his sons and grandsons and his daughters and granddaughters—all his offspring. Genesis 46:5-7

Though not by name, the women are mentioned in summary form: The wives of Jacob's sons, and his daughters and grandaughters. No wife of Jacob is mentioned in the list of those who moved to Egypt. It would be an incredible insult if the sons of Jacob had "forgotten" to take their mother with them, and Jacob had "forgotten" to remind them. The statement "Israel's sons took their father Jacob ...." indicates that Jacob had become old and frail. According to Genesis 47:8-9, Jacob was 130 years old at that time. This also makes it highly unlikely that any of his wives would still be alive.

Finally, feeling his death approach some time later (in Egypt), Jacob calls all his sons to himself:

Then he gave them these instructions: "I am about to be gathered to my people. Bury me with my fathers in the cave in the field of Ephron the Hittite, the cave in the field of Machpelah, near Mamre in Canaan, which Abraham bought as a burial place from Ephron the Hittite, along with the field. There Abraham and his wife Sarah were buried, there Isaac and his wife Rebekah were buried, and there I buried Leah. The field and the cave in it were bought from the Hittites." Genesis 49:29-32

This clearly states that Jacob buried Leah in Canaan. Given the frailty and age of Jacob at the time of his arrival in Egypt, he hardly went back to bury her there after they had immigrated to Egypt. No, this happened before they left Canaan. Leah was buried where she died, just as Rachel had been buried where she had died (Genesis 35:19-20).

Conclusion: Joseph's mother died long before Joseph himself came to Egypt. Leah also died before Jacob and his household moved to Egypt. There was no wife of Jacob that moved with him. The Biblical data are clear. When speaking about Joseph's parents coming to Egypt, the Qur'an commits a clear error.