József Müllner was born on June 24, 1899, in Káld. He lost his parents during his childhood. According to an Argentine biographical account, he trained as a master carpenter and began working at a young age. He was only able to pursue his calling to the priesthood later: in 1930, he was accepted into the seminary of the Archdiocese of Esztergom.

He was sent to South America to provide pastoral care for Hungarian immigrants in Argentina. He arrived in Buenos Aires on August 24, 1936, where he completed his theological studies at the Metropolitan Seminary in Villa Devoto. He was ordained a priest in Buenos Aires on September 18, 1937.

He began his priestly ministry in the Diocese of Mendoza. He served as episcopal secretary and hospital chaplain, and was officially listed among the employees of the Diocese of Mendoza at least as early as 1950. He organized a medical clinic to support the poor and the sick, distributed food to those in need, and used donations to build the Virgen del Valle Chapel.

The Colegio Padre José Müllner in Godoy Cruz still bears his name today; it is listed in the official educational registry of Mendoza Province under institutional code P-082 at 2507 Talcahuano Street.

He passed away unexpectedly in Argentina on February 14, 1961. His estate was auctioned off, and the proceeds were sent to his impoverished relatives living in Hungary. According to a memorial written about him, he always remained conscious of his Hungarian heritage and supported his Hungarian compatriots during his time in Argentina.