Máté Fülep was born in Miskolc, where at age 14, at the recommendation of his first voice teacher, he started attending the city's Béla Bartók High School of Music, graduating in solo voice and double bass. During his high school years, he received regular tutelage from Boldizsár Keönch. In 2004, he was admitted into the University of Debrecen's Faculty of Music, where his teachers were Éva Szerdahelyi and Judit Böszörményi. He made his operatic debut in 2009, in a production of Le nozze di Figaro at the National Theatre of Szeged. After posting outstanding results at competitions, he started to receive more and more solo roles at Debrecen's Csokonai National Theatre, of which the most important were Ping (Puccini's Turandot), Shchelkalov (Mussorgsky's Boris Godunov) and Morales and Dancairo (Bizet's Carmen).
In 2009 he won the Scholarship of the Republic of Hungary and graduated from the university with honours with a degree in voice and voice pedagogy. He took the stage of the National Theatre of Miskolc as Silvio in Pagliacci and as Marcello in La bohème. In 2010, he won a scholarship from the Wagner Society, and continued his studies at the Royal Conservatory of The Hague under Andrew Schroeder and Rita Dams. He sang in numerous oratorios and song recitals in the Netherlands, thus expanding his vocal repertoire. 2011 saw him sing the title role of Don Giovanni at The Hague's Schouwburg. He has worked with such conductors as Gijs Leenaars, Marc Albercht, Stefano Montanari and Yannick Nézet-Seguin.
In 2013 he reached the finals of the Eighth Hilde Zade Singing Competition in Vienna, and later that year sang the role of Frank/Fritz in Korngold's Die tote Stadt at the Bartók Plus Opera Festival in Miskolc. Winning the Annie Fischer Scholarship for the Performing Arts in 2014 led to numerous opportunities to make appearances in Hungary, at which he primarily sang songs. Most noteworthy of these were Schubert's Winterreise song cycle and László Lajtha's settings of folk songs.
At Debrecen's Csokonai National Theatre during the 2015/16 season, he sang the role of Belcore in L'elisir d'amore and Baron Douphol in La traviata. He was invited to make his debut at the Hungarian State Opera in September of 2015, as Eustachio in Donizetti's I pazzi per progetto.