The Koran alludes to or retells (with varying degrees of accuracy) many of the stories of the Old Testament. It also presumes to give additional details to many of the stories. In one of them, for instance, Allah enables Solomon to understand the speech of birds and ants (Sura 27:16ff.).
There are allusions to the New Testament also. Jesus is mentioned numerous times. He is said to be a virgin-born prophet and a worker of miracles, but he is emphatically not the Son of God.
It is doubtful that Muhammad ever had the opportunity to know orthodox Christians or to read from the New Testament itself. Many of the stories about Jesus that appear in the Koran are derived from the writings of heretical Christian groups. For instance, the Koran has the boy Jesus fashion birds of clay and breathe into them the breath of life so that they become living creatures. This is taken from the second-century apocryphal Gospel of Thomas.
Spencer gives numerous other examples in chapter 3.