No, Moslems don’t believe that Jesus was the Messiah.
Think of it like a movie. The Torah is the first one, and the New Testament the sequel. Then the Qu’ran comes out, and it retcons the last one like it never happened. There’s still Jesus, but he’s not the main character anymore, and the messiah hasn’t shown up yet.
Jews like the first movie but ignored the sequels. Christians think you need to watch the first two, but the third movie doesn’t count. The Moslems think the third one was the best, and Mormons liked the second one so much, they started writing fanfiction that doesn’t fit with ANY of the series canon.