The Vatican announced the appointment Jan. 18.
After serving as vicar general of the Diocese of Derry, Northern Ireland, last year Archbishop-designate Martin became its diocesan administrator upon the retirement of Bishop Seamus Hegarty.
In brief remarks to the media in Armagh, Archbishop-designate Martin said, "There is need for renewal in the church, so that the message of Christ, in all its richness, is presented in ways which engage a new generation.
"There is a need for a mature relationship between church and society, in both parts of this island, and people of faith have a vital role to play.
"It would hugely impoverish our faith if we were expected to 'leave it at home' or 'keep it for Sundays,' excluding it from our conversations and actions in daily life," he said.
He also said one of the biggest challenges facing the Catholic Church "is to acknowledge, live with, and learn from the past, including the terrible trauma caused by abuse."
Cardinal Brady, 73, has served as primate of all Ireland and president of the Irish Episcopal Conference since 1996, and his tenure has been marked by controversy over clerical sex abuse. Advocates for abuse victims have called for his resignation since 2010, when it emerged that he had failed to inform civil authorities about an abusive priest who went on to molest children in several countries.
The cardinal will turn 75, the age at which a bishop must submit his resignation to the pope, in August 2014. As coadjutor, Archbishop-designate Martin will help run the Armagh Archdiocese until the cardinal's retirement.
The archbishop-designate, 51, was born in Londonderry, Northern Ireland, and attended the seminary at the Pontifical University at St. Patrick's College, Maynooth. He holds a master's degree in educational administration from the University of Cambridge, England.
Archbishop-designate Martin has served as president of St. Columb's College in the Diocese of Derry, and executive secretary of the Irish bishops' conference.
According to a statement from the bishops' conference, the archbishop-designate has made frequent appearances on devotional television programs and has a "particular interest in sacred music, especially Gregorian chant."