Montes was born in West Germany, where her father, Alberto Montes, was posted as a United States Army doctor. Her family was of Asturian origins (a region in Spain), and her grandparents had immigrated to Puerto Rico. The family later lived in Topeka, Kansas, and then Towson, Maryland, where she graduated from Loch Raven High School in 1975. In 1979 she earned a degree in foreign affairs from the University of Virginia, and in 1988 she finished a master's degree at Johns Hopkins University School of Advanced International Studies.