Vólquez signed with the Rangers in 2001 under the name of Julio Reyes. He went by Edison Vólquez after 2003, before adding an n to his first name in 2007.

Vólquez grew up in the Dominican Republic and started playing baseball when he was 9 or 10 with the support of his parents. "It was good for me because my mom and dad always took care of me... The only thing I did was go to school and play baseball."

Vólquez was signed as an amateur free agent by the Texas Rangers in the Dominican Republic in 2001. Together with John Danks and Thomas Diamond, Vólquez was one third of the "DVD" trio of Rangers pitching prospects.

After spending four years in the Rangers' minor league system, Vólquez made his Major League debut on August 30, 2005 in a start against the Chicago White Sox. He lost all three Major League games he started that season, as well as one of the three games in which he appeared as a reliever, and posted a 14.21 ERA. He spent the first five months of the 2006 season with the Oklahoma RedHawks of the Class AAA Pacific Coast League until he was recalled to the majors in September. This time, he fared better, winning one of his eight starts and posting a 7.29 ERA.

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