[1] Quevedo was appointed Bishop-Prelate of Kidapawan by Pope John Paul II on 28 October 1980.

[2] On 12 January 2014, Pope Francis named Quevedo as one of 19 men to be inducted into the College of Cardinals in the consistory of 22 February 2014, with the titular church of Santa Maria "Regina Mundi" a Torre Spaccata.

[3][4] He is the first cardinal from Mindanao[4] and, until his 80th birthday, the second Filipino cardinal-elector, along with Luis Antonio Tagle, Archbishop of Manila.

Due to travel restriction due to COVID-19 Pandemic and Apostolic Nuncio to the Philippines is vacant, Cardinal Quevedo officially installed Bishop Charlie M. Inzon as sixth Apostolic Vicar of Jolo on April 4, 2020 and Archbishop Jose Cabantan as Archbishop of Cagayan de Oro on August 28, 2020.

In 1994 Quevedo received the most votes for election to the General Council of the Secretariat of the Synod of Bishops in Rome.