Short answer
The Quran does not name it. It says every messenger was sent in the language of his people, and Jesus was sent to the Children of Israel, so he spoke their language. Historians generally hold that this was Aramaic, with Hebrew used for scripture. Jesus did not speak Arabic as his own language.
There is no fiqh ruling here. Muslims affirm what the Quran states, that Jesus spoke the language of his people, and leave the details to history. Knowing the name of his language is not part of belief.
The four schools agree on this.
Explanation
The Quran sets the rule for every prophet: Allah sent no messenger except in the language of his people, so that he could make the message clear to them. Jesus was a messenger to the Children of Israel, so he preached to them in their own tongue. The Injil (Gospel) revealed to him would have been in that language too.
Historians say the everyday language of Jews in Galilee and Judea in the first century was Aramaic, while Hebrew was used in scripture and worship, and Greek was common in the wider Roman east. Islamic sources do not contradict this; they simply do not go into it.
His name follows the same pattern. Muslims know him by the Quranic Arabic name Isa, while English uses Jesus, from Greek and Latin forms of his Hebrew or Aramaic name. See why Muslims call Jesus Isa.
One special moment of speech the Quran does record: when Mary brought the baby Jesus to her people and they accused her, she pointed to him, and he spoke from the cradle: I am the servant of Allah. He has given me the Scripture and made me a prophet. That miracle, not the language, is what the Quran wants us to remember. See the miracles of Jesus.
The evidence
Quran, Surah Ibrahim 14:4
وَمَآ أَرْسَلْنَا مِن رَّسُولٍ إِلَّا بِلِسَانِ قَوْمِهِۦ لِيُبَيِّنَ لَهُمْ ۖ فَيُضِلُّ ٱللَّهُ مَن يَشَآءُ وَيَهْدِى مَن يَشَآءُ ۚ وَهُوَ ٱلْعَزِيزُ ٱلْحَكِيمُ ﴿٤﴾
And We did not send any messenger except [speaking] in the language of his people to state clearly for them, and Allah sends astray [thereby] whom He wills and guides whom He wills. And He is the Exalted in Might, the Wise.
Every prophet spoke the language of the people he was sent to.
Quran, Surah Aal-i-Imraan 3:49
وَرَسُولًا إِلَىٰ بَنِىٓ إِسْرَٰٓءِيلَ أَنِّى قَدْ جِئْتُكُم بِـَٔايَةٍۢ مِّن رَّبِّكُمْ ۖ أَنِّىٓ أَخْلُقُ لَكُم مِّنَ ٱلطِّينِ كَهَيْـَٔةِ ٱلطَّيْرِ فَأَنفُخُ فِيهِ فَيَكُونُ طَيْرًۢا بِإِذْنِ ٱللَّهِ ۖ وَأُبْرِئُ ٱلْأَكْمَهَ وَٱلْأَبْرَصَ وَأُحْىِ ٱلْمَوْتَىٰ بِإِذْنِ ٱللَّهِ ۖ وَأُنَبِّئُكُم بِمَا تَأْكُلُونَ وَمَا تَدَّخِرُونَ فِى بُيُوتِكُمْ ۚ إِنَّ فِى ذَٰلِكَ لَءَايَةًۭ لَّكُمْ إِن كُنتُم مُّؤْمِنِينَ ﴿٤٩﴾
And [make him] a messenger to the Children of Israel, [who will say], 'Indeed I have come to you with a sign from your Lord in that I design for you from clay [that which is] like the form of a bird, then I breathe into it and it becomes a bird by permission of Allah. And I cure the blind and the leper, and I give life to the dead - by permission of Allah. And I inform you of what you eat and what you store in your houses. Indeed in that is a sign for you, if you are believers.
Jesus was sent to the Children of Israel.
Quran, Surah Maryam 19:29-30
فَأَشَارَتْ إِلَيْهِ ۖ قَالُوا۟ كَيْفَ نُكَلِّمُ مَن كَانَ فِى ٱلْمَهْدِ صَبِيًّۭا ﴿٢٩﴾ قَالَ إِنِّى عَبْدُ ٱللَّهِ ءَاتَىٰنِىَ ٱلْكِتَٰبَ وَجَعَلَنِى نَبِيًّۭا ﴿٣٠﴾
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.
Jesus spoke miraculously as a baby.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.