In 1976, Quivar portrayed Serena in the Cleveland Orchestra's production of Gershwin's Porgy and Bess. The concert was recorded and went on to win a Grammy Award for best opera recording. The following year Quivar made her debut at the Tanglewood Festival singing in the world premiere of Roger Sessions When Lilacs Last In The Dooryard Bloom'd with the Boston Symphony Orchestra. She also made her Metropolitan Opera début on October 10, 1977 as Marina in Boris Godunov. She became a regular at the Met during the 1980s and 1990s, appearing as Jocasta in Oedipus rex, Suzuki in Madama Butterfly, Isabella in L'italiana in Algeri, Federica in Luisa Miller, Fidès in Le prophète, Frugola in Il tabarro, Mother Marie in Dialogues des Carmélites, Louis XV Chair in L'enfant et les sortilèges, the Princess in Suor Angelica, Ulrica in Un ballo in maschera, and Serena in Porgy and Bess. Her 101st and last performance at the Met was in a concert performance of Verdi's Requiem in 1997 where she sang the mezzo soprano solos under the baton of James Levine.
Quivar's other opera credits include performances at the Deutsche Oper Berlin, Bavarian State Opera, La Scala, Teatro la Fenice, Teatro dell'Opera di Roma, the Theatro Municipal in Rio de Janeiro, Teatro Colón, Royal Opera at Covent Garden, Houston Grand Opera, Seattle Opera, and Los Angeles Opera among others. Her other roles includes Adalgisa in Norma, the title role in Carmen, Erda in Siegfried and Das Rheingold, Brangäne in Tristan und Isolde and Orpheus in Gluck's Orfeo ed Euridice, the latter being a role with which she became particularly associated.
She has also performed with many of the world's premiere orchestras, including the New York Philharmonic, Los Angeles Philharmonic, Cincinnati Symphony, Chicago Symphony Orchestra, Philadelphia Orchestra, Cleveland Orchestra, San Francisco Symphony, Toronto Symphony, Orchestre de Paris, Berlin Philharmonic, London Philharmonic, Montreal Symphony and the BBC Symphony Orchestra to name just a few.
Quivar has taken on the task of rescuing the works of forgotten composers, concentrating on those of African-American composers of the 19th and 20th centuries. Her stated goal is "to compile a program of these neglected composers and someday record them." She has also performed in productions of African-American composers' works, as well as a 1981 revival of Virgil Thomson's Four Saints.
She has also been a champion of new music. In 1999 she performed the role of The Goddess of the Waters in the world premiere of Anthony Davis' opera Amistad at the Lyric Opera of Chicago. She also premiered William Bolcom's song cycle From the Diary of Sally Hemmings at the Library of Congress in 2001. She has since performed the cycle in recitals throughout the United States in a tour with Harolyn Blackwell in 2002-2003.
A native of Philadelphia, Ms. Quivar is a graduate of the Philadelphia Academy of Music and a former member of the Juilliard Opera Theatre. She joined the Academy of Vocal Arts faculty in 2014.