Does a Jew convert to Christianity? No. Judaism is Christianity. Christianity is not Judaism. Christians are to hold to Judaism as Jesus and the Apostles uphold Judaism. The Jew has to discover YHOSHUA in the Torah as YHWH in flesh and believe (John 3:16) to be saved (Romans 10:9-11).
(17) "Don't think that I have come to abolish the Torah or the Prophets. I have come not to abolish but to complete. (18) Yes indeed! I tell you that until heaven and earth pass away, not so much as a yud ( ' ) or a stroke will pass from the Torah—not until everything that must happen has happened.
Matthew 5:17-18 CJB
But it is easier for heaven and earth to pass away than for one stroke of a letter in the Torah to become void. Luke 16:17 CJB
(29) But Avraham said, 'They have Moshe and the Prophets; they should listen to them.' (30) However, he said, 'No, father Avraham, they need more. If someone from the dead goes to them, they'll repent!' (31) But he replied, 'If they won't listen to Moshe and the Prophets, they won't be convinced even if someone rises from the dead!' Luke 16:29-31 CJB