Firstly, the idea that Quetzalcoatl is, in fact, Jesus Christ, is not official LDS Church doctrine, and neither does the comparison feature particularly critically in the faith of the typical Mormon, as I understand it. It's apologia, not dogma.
Which isn't to say the idea can be dismissed there. It seems to be a reasonably popular position in Mormon apologetics. Indeed, the third president of the LDS church, John Taylor, was fairly certain of the link:
The story of the life of the Mexican divinity, Quetzalcoatl, closely resembles that of the Savior; so closely, indeed, that we can come to no other conclusion than that Quetzalcoatl and Christ are the same being.
(That still doesn't establish the position as church doctrine, since prophets are fallible)
What this is attempting to establish is: The Book of Mormon writes that Jesus appeared on the American continent, in 3 Nephi. Quetzalcoatl has proved a compelling myth to attach the story to, and suggest this is the native record of the life of Jesus.
Some of the parallels identified are as follows (sources: 1, 2):
- Identity as a creator deity
- Depiction as white, with a long beard and european features
- Virgin Birth
- Teacher of virtue
- Bringer of sustenance ("Bread of Life")
- Sacrifice for the benefit of man
- Hanging from a "tree"
- Belief in a later return
Which is all fairly compelling. The trick is figuring out which dates to the pre-Columbian era, and which resemble Christian tradition because they were directly influenced by the Christian (ie. Spanish) conquest and colonization. As Diane E. Wirth states in her paper "Quetzalcoatl, the Maya Maize God, and Jesus Christ":
...Spanish chroniclers, desiring to please adherents of both Christianity and the religion of the indigenous natives, emphasized the powerful symbolic continuity between the Catholic and Mesoamerican belief systems. They did this by frequently combining myth and history from pre-Hispanic times. Such manipulation was even a native tradition in Mesoamerica. Kings caused historical records to be manipulated in order to strengthen and authenticate their legitimacy to rule their people. Because of these practices, scholars are sometimes in a quandary as to what is historical and what is mythological.