Márquez signed with the Chicago Cubs as an international free agent, in August 2015. He made his professional debut in 2016 with the Dominican Summer League Cubs, going 4–2 with a 1.48 earned run average (ERA) over 12 starts. Márquez played 2017 with the Arizona League Cubs, pitching to a 2–1 record and a 5.52 ERA over 11 games (nine starts), and 2018 with the Eugene Emeralds and South Bend Cubs, compiling a 1–4 record with a 3.13 ERA over 12 starts.
Márquez started 2019 with South Bend before being promoted to the Myrtle Beach Pelicans. Over 22 starts between the two teams, he pitched to a 9–5 record with a 3.13 ERA, striking out 128 over 103+2⁄3 innings.
Márquez was selected to the 40-man and active rosters by the Cubs on September 27, 2020. He made his MLB debut that night against the Chicago White Sox, pitching 2⁄3 of an inning of relief.
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change-up grip