Yes, Christianity was born out of Judaism. Jesus was, as far as we know, born in Bethlehem, near Jerusalem, and was crucified in Jerusalem and on the third day rose from the dead.
Judaism may not have been the first monotheistic religion in the Near East - Zoroastrianism may have a valid claim to that distinction - but it remains monotheistic, as does Christianity. The scriptures of the Jewish religion, known to Jews as the Tanach, became scriptures of Christianity, known as the Old Testament, to which were added the Christian New Testament scriptures. Although Jesus did not actually say that Christians should use the scriptures that now form the Old Testament, he did say that he came to fulfil the law of the Jews.
The Judaism that Jesus knew was not the rabbinic Judaism we know today. What we call Second Temple Judaism ended in 70 CE with the destruction of the great temple in Jerusalem. Christianity and rabbinic Judaism both came out of Second Temple Judaism, and each offered a new way forward, for those who believed in God. John Dominic Crossan says, in the Prologue to The Birth of Christianity, Christianity and rabbinic Judaism each claimed exclusive continuity with the past, but in truth each was as great a leap and as valid a development from that common ancestry as was the other. Crossan describes Christianity and rabbinic Judaism as like two children of the same mother.