Márquez made his American debut the following year with the Rookie-league Princeton Rays, logging a 2-5 record and 4.05 ERA with 38 strikeouts in 12 starts.

In 2014, Márquez pitched for the Single-A Bowling Green Hot Rods, logging a 5-7 record and 3.21 ERA with 95 strikeouts across 22 games (18 starts).

In 2015, he moved up a level to the High-A Charlotte Stone Crabs, registering a 7–13 record and 3.56 ERA with 104 strikeouts across 26 appearances (23 starts).

In his time in the Rays system, Márquez pitched primarily as a starter, with a cumulative 14–27 record, 3.88 ERA, and 266 strikeouts in 324+2⁄3 innings.

[3] On January 28, 2016, the Rays traded Márquez and reliever Jake McGee to the Colorado Rockies in exchange for outfielder Corey Dickerson and infield prospect Kevin Padlo.

[4] Márquez started 2016 with the Double-A Hartford Yard Goats and was promoted to the Triple-A Albuquerque Isotopes in early August.

[5] After three relief appearances, he won his first MLB start, allowing one run in five innings in an 11–1 win over the St. Louis Cardinals.

[11] On July 11, 2018, Márquez hit his first career home run off second baseman and former teammate Daniel Descalso in a blowout 19–2 win over the Arizona Diamondbacks.

[13] On September 26, Márquez tied the modern-day MLB record by striking out the first eight batters of a game against the Philadelphia Phillies, before pitcher Nick Pivetta reached base on an error on a 0–2 pitch.

[16][17] Márquez also had a phenomenal season at the plate for a pitcher, hitting .300 with a home run, 5 RBIs, and a .650 on-base plus slugging.

Marquez's performance, along with fellow second-year starter Kyle Freeland, helped the Rockies reach the postseason in back-to-back years for the first time in franchise history.

Márquez started the NL West tie-breaker game, picking up loss against the Los Angeles Dodgers after allowing two-run home runs to Cody Bellinger and Max Muncy in 4+2⁄3 innings.

[20] On April 14, 2019, Márquez threw the first one-hit shutout in Rockies franchise history in a 4–0 win, yielding only a single to Evan Longoria in the 8th inning.

Márquez hit his second career MLB home run on July 31, against Yu Darvish of the San Diego Padres.

[38] On August 8, manager Bud Black announced that Márquez would miss the remainder of the season after suffering a stress reaction in his elbow.