The Book of Mormon does not have a "story of Christmas" in the same sense as the Bible does, because it was not set in the land of Israel (except for the very beginning of the book.) Therefore, there is no coverage of the birth of the baby Jesus and the events surrounding it.
It does, however, repeatedly prophesy of the coming of the Savior, that he would be born at Jerusalem and that he would be born of a virgin. Also, in Helaman chapter 14, we have a prophecy of the birth of Christ, and also of his death, and of the signs that would be given. Interestingly, these signs are consistent with the Biblical events mentioned, but also much greater than them. At Jesus's birth, a new star appeared in the sky, and also the sky remained as light as day throughout the night, even though everyone saw the sun go down. And when Christ died, there were not only earthquakes, but immense devastation throughout the land, consistent with a major volcanic eruption.
The bit about Christ being born at Jerusalem is very interesting, since the Bible clearly states that he was born in Bethlehem, a small town a ways away from the great city. While at first glance this appears to be an obvious error in the Book of Mormon, modern scholarship has shown the exact opposite: the non-Biblical phrase "the land of Jerusalem" used in the Book of Mormon was an authentic term used to describe the entire region, as shown by writings found in both the Dead Sea Scrolls and the Tell el Amarna letters.