His medical career spanned 50 years in his native Bavaria, and included being a lecturer, teacher and consultant for orthopedic surgery, and physical therapy.

He worked at hospitals in Vilsbiburg, Rosenheim, and Murnau am Staffelsee, and practiced as a sports doctor in Munich and Weilheim from 1956 to 1990.

He also served as an assistant to Fritz Lange at the Klinikum Harlaching [de], part of the University of Munich.

[1] At the national level, Montag was involved in the renaming of the German Orthopedic Society (Deutsche Orthopädische Gesellschaft) into the German Society for Orthopedic Traumatological Sports Medicine (Deutsche Gesellschaft für Orthopädisch-Traumatologische Sportmedizin) in 1962.

[3] He presided over the 1983 European Figure Skating Championships hosted in Dortmund, and served as vice-president of the German Ice Sports Federation in 1988.

He advised against starting the races over 1,500 metres (4,900 ft) in distance at the outdoor James B. Sheffield Olympic Skating Rink in 1980, when temperatures dropped below −20 °C (−4 °F) and exposed athletes to frostbite.

[3] From 1975 to 1998, Montag served as the Chief Medical Officer (CMO) of the International Ice Hockey Federation (IIHF).

[5][8][9] In 1975, Montag received an honorary diploma from the IIHF, and the golden badge of honor with diamonds from the German Ice Sport Federation.

[6] He was honored by the French Minister of Youth and Sport in 1985, with the Ordre du Mérite sportif of France.

[6] He was made an honorary member of the JOSSM in 1988,[6] and received honors from the Western Pacific Orthopedic Association in 1989.

[4] He was the inaugural recipient of the Paul Loicq Award in 1998, in recognition of "outstanding contributions to the IIHF and international ice hockey".