Short answer is YES they do.
Muslims have always believed in the Prophet Jesus (as), in fact they believe in other Prophets as well, many mentioned in the Quran like Adam, Noah (Nowh), Abraham (Ibrahim), David (Dawud), John (Yehya), Moses (Musa) (as) and the list goes on. There is no distinction between these prophets and they are all considered messengers who came with one main core message to their people.
That message is Strict Monotheism -Belief in one God- and not associating any partners with God.
Jesus in Islam
Jesus is only a prophet in Islam, Muslims don’t believe in the Trinity and do not take Jesus as God nor a son of God nor a father. They do not believe that Mary (Maryam) is the mother of God, rather they believe that Mary gave birth to Jesus miraculously. Muslims do not believe that Jesus was crucified nor died on the cross, rather someone was made to like like him, so he was saved and lifted to the heavens. Muslims believe that Jesus will come down at the end of times.
Islam and the past scriptures
As part of Islam, Muslims acknowledge the Past Scriptures that the prophets came with such as the Psalms, Gospel and Torah etc. and the corresponding prophets that brought these scriptures. Muslims believe that all past scriptures with the exception of the Quran have been re-written and modified over time, to the point where they now carry some truth mixed with falsehood.
Acknowledge here means that Islam tells us that the previous prophets came with scriptures and teachings that were accurate at the time and only relevant to the nation or people that lived around the time of that corresponding prophet.
What do muslims believe and follow ?
Muslims follow the Quran revealed to Prophet Muhammed (Pbuh) which is the continuation of the same message that Adam, Abraham, Moses, Jesus and other prophets called to; That is strict Monotheism.
Monotheism is the belief that their is only one God worthy of worship. God does not have any partners nor a son.
Muslims follow the Quran because God has told us to. The Quran continues that core message of strict monotheism with teachings relevant for all of humanity to live their life till the end of Times. Prophet Muhammed(pbuh) is the Last and Final Messenger and no other new Prophet will come after him. Muslims do not worship Prophet Muhammed (pbuh), rather they Worship God and follow the prophets teachings which were inspired by God.
Still not convinced? Keep on Asking questions. Islam will not leave them unanswered, their are always answers backed by evidence and not feelings and emotions.
Say, He is God, the One. God, the Absolute. He does not give birth, nor was He born, and there is nothing like Him. Quran [112: Verses 1-4]
Below are a few Links for further reading. You deserve to know the truth and receive closure. Feel free to reach it out if you have further Questions or simply do your own rigorous research from reputable sources.
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Further Reading Links
Jesus as
25 Verses where Jesus and Adam were mentioned in Quran
Read the Quran Online
A verse from the Old Testament confirming that God is One — A man came to Jesus and asked ‘Which is the first commandment of all?’ Jesus answered: The first of all the commandments is Hear, O Israel, the Lord our God, the Lord is one. [Mark 12:28-29]