McQuaid was raised in Jersey City, then in the Diocese of New York. Ordained in 1848, became cathedral rector and then vicar-general of Newark after that diocese was formed in 1853. He also assisted the bishop in the founding of Seton Hall College. When the Diocese of Rochester was created in 1868, he was named its first bishop. He presided over enormous institutional expansion during his long reign there, and became nationally known as one of the opponents of the Americanists in the controversy that roiled the American Church in the last decades of the nineteenth century.