The source of all the information about Jesus is found in the NT. That's where
the founder of Islam would find the information, find out about Jesus.

In the bible, in the NT is where we learn about the crucifixion, about
the 'thief on the cross', about the Roman soldiers guarding the tomb,
about Pilate and Herod and all those historical people of that time.

Why say they believe about Jesus at ALL, if they don't believe in what
He said and did.

Because they didn't check to make sure the Quran was the true word of God the way the Christians checked to make sure their Bible was true.

Logically, God would never allow His own word to become corrupted. But instead of checking things out first, they just believed what they were taught.

Also, a lot of their teachers tell them things that aren't true about the Bible - but if they checked these facts with even secular sources, they would find out that the Quran is actually false.