It is no doubt nothing more than a coincidence that the first two times (1928 and 1960) the Democrats nominated a Roman Catholic for president (Al Smith and John F. Kennedy) he was opposed by a Republican Quaker (Herbert Hoover and Richard Nixon). Still, the fact was occasionally remarked upon in 1960. North Carolina Governor Luther Hodges (later to become JFK's Secretary of Commerce) reminded an audience in Virginia that in 1928 voters had chosen a Quaker over a Catholic and had lived to regret it, and that "if you vote for a Quaker this time, you'll regret it horribly."
https://books.google.com/books?id=V5aZBgAAQBAJ&pg=PA96
Challenge: what are other cases in which one party might plausibly nominate a Catholic and the other a Quaker?
(1) 1920: The Democrats could plausibly have nominated the "fighting Quaker" A. Mitchell Palmer for President, but it's hard to think of a plausible Republican Catholic nominee. (Maybe Charles Bonaparte, though he didn't have long to live?
https://en.wikipedia.org/wiki/Charles_Joseph_Bonaparte Very unlikely, but you can never tell what a deadlocked convention will do...)
(2) 1920: Two deadlocked conventions produce two surprise choices--Pennsylvania Governor William Sproul
https://en.wikipedia.org/wiki/William_Cameron_Sproul for the Republicans and New York Governor Al Smith for the Democrats. (Yes, I know it's probably too early for Al Smith.) Or Hoover just might have gotten the GOP nomination in 1920 instead of 1928. (He was of course also spoken of as a possible Democratic candidate.)
(3) "In 1920, Sproul was offered the nomination to become the presidential running mate of Warren G. Harding, but he declined. Had Sproul accepted a role as vice president, it is probable he would have become president of the United States assuming Harding had still died in office. Instead, Calvin Coolidge stepped into that role."
http://www.phmc.state.pa.us/portal/communities/governors/1876-1951/william-sproul.html So 1924 witnesses Sproul versus Al Smith or Thomas J. Walsh.
https://en.wikipedia.org/wiki/Thomas_J._Walsh (Walsh, who was chairman of the Democratic national convention in 1924, started to be seriously spoken of as a compromise nominee as the deadlock dragged on. McAdoo indicated a willingness to accept him because, though a Catholic, he was a westerner, an economic progressive, a supporter of Prohibition, and untied to urban machines. But Walsh made it clear he would not be a candidate. )
(4) Senator Arthur Capper of Kansas
https://en.wikipedia.org/wiki/Arthur_Capper was a Quaker. I suppose that if something happened to Hoover, Capper could have been the GOP nominee against Smith or Walsh in 1928.
(5) Or maybe Coolidge chooses California governor Friend Richardson
https://en.wikipedia.org/wiki/Friend_Richardson as his running mate in 1924 and Richardson is the GOP presidential nominee in 1928. (I like the idea of a Quaker nominee named "Friend"...)
(6) 1952: Paul Douglas
https://en.wikipedia.org/wiki/Paul_Douglas vs. Joe McCarthy! (OK, now I'm really stretching things...)
(7) A lot of things would have to change but a 2016 race between John Hickenlooper
https://en.wikipedia.org/wiki/John_Hickenlooper and Paul Ryan is conceivable.
(In 2004 the Democrats finally nominated a Catholic again--John Kerry. But I don't see GW Bush converting to Quakerism...)