The morion style helmet became popular in the early 1500s and was widely used by the Spanish in Europe as well as in the New World.
It is probably anachronistic to apply it to the early conquistadors such as Cortez. It had not become common at that time. Even Pizarro in 1532 would have probably been too early for his men to have been equipped with morion helmets.
Later Conquistadors such as DeSoto or Coronado would have more likely had their men equipped with morions.
http://latinamericanhistory.about.com/od/theconquest/a/09armsconquest_2.htm