Márquez was born in Álamos, Sonora, in 1950 where his interest in music began.
Although Márquez was already an accomplished composer in Mexico, his music started to reach the international stage with the introduction of his series of Danzones in the early 1990s. The Danzones are based on the music of Cuba and the Veracruz region of Mexico. In particular, Danzón no. 2 has received international acclaim and has catapulted the popularity of the composer. Danzón no. 2 was commissioned by the National Autonomous University of Mexico (UNAM) and was debuted in 1994 in Mexico City by the Orchestra Filarmonica de la UNAM under the direction of Francisco Savin. Due to the popularity of this piece, Danzón no. 2 has been nicknamed, along with Huapango by Pablo Moncayo, as the second national anthem by the Mexican people. Danzon no. 2 was included on the program of the Simon Bolivar Youth Orchestra conducted by Gustavo Dudamel on their 2007 tour of Europe and the United States. As a result of the strong public response to the orchestra's performance of the piece, Danzon no. 2 has established itself as one of the signature pieces performed by the orchestra. It has also opened the door for the discovery of other pieces by the composer that are increasingly being performed throughout the world and extensively in Latin America. Son a Tamayo for harp, percussion and tape was featured at the 1996 World Harp Congress.