[3] Treviño's conducting mentors have included Leif Segerstam, Kurt Masur, Michael Tilson Thomas, and David Zinman.

[5] In 2010, Treviño won the James Conlon Prize for Excellence in Conducting at the Aspen Music Festival and School.

[12][13] In June 2022, the Basque National Orchestra announced the extension of his music directorship for an indefinite tenure.

[14] In January 2025, the board of the Basque National Orchestra terminated Treviño's contract as music director with immediate effect.

[2] In May 2021, the Malmö Symphony Orchestra announced that Treviño is to stand down as its chief conductor at the close of the 2020-2021 season, and then to take the title of artistic adviser for 2 years.