No - none of the accounts of the Massacre of the Innocents suggest John the Baptist died in it (it would be odd if he had, as then he would not be able to baptise Jesus later in life).
The Massacre of the Innocents is at
Matthew 2:16-18, but does not record any reference to John the Baptist.
The story is not mentioned by the contemporary Jewish historian Josephus, nor in the other gospels, nor in the early apocrypha. Its first appearance in any source other than Matthew being the 2nd century being in the Protoevangelium of James of c.150 AD, which excludes the Flight into Egypt and switches the attention of the story to the infant John the Baptist. In relation to John, it states:
- "And when Herod knew that he had been mocked by the Magi, in a rage he sent murderers, saying to them: Slay the children from two years old and under. And Mary, having heard that the children were being killed, was afraid, and took the infant and swaddled Him, and put Him into an ox-stall. And Elizabeth, having heard that they were searching for John, took him and went up into the hill-country, and kept looking where to conceal him. And there was no place of concealment. And Elizabeth, groaning with a loud voice, says: O mountain of God, receive mother and child. And immediately the mountain was cleft, and received her. And a light shone about them, for an angel of the Lord was with them, watching over them."
Most recent biographers of Herod therefore do not regard the massacre as an actual historical event, but rather, like the other nativity stories, as creative hagiography. The gospel of Matthew was most likely written circa 80 - 85 AD by an anonymous Christian appealing to a Jewish audience. The author's goal was to portray Jesus as the Jewish Messiah, the fulfillment of Jewish prophecy, and a greater Moses.