the only time i've seen them do it without bribes is as Rancid Sushi said, when they're preparing to beat someone to bits.
some of them are open to bribes, if you have enough to bribe them with (techs/cash/good relations to start with), and if FR isn't their favorite civic in the first place.
"I'm asking because I've got this game going where I'm aiming for a diplo win which hinges on the support of another civ that (as of now) shares my religion."
good thinking. i also sometimes bribe Leader X out of a civic that's earning them +wise civics with Leader B when X is my UN competition and i want B to vote for me *giggle*. for getting them adopt a religion again, spreading multiple religions to their cities can encourage them if you have the ability to do that.
yeah i just opened a save to check. culture game but for me those are 90% diplomacy anyway (my army is like 12 troops). right now i'm using no state religion for political reasons, but i'm running organized religion, for +wise civics. washington is pleased with me. he's using FR, but will adopt OR for 910g. i obviously can't ask him to convert to a specific religion (i forget if he even has one that i have), but i can get him to use a civic that will force him back to the religion he was using at the time he changed to FR. churchill, exactly the same. pleased, will do it for 910g. mansa, friendly, but redlined for "that would go against everything we stand for". freddy, pleased, redlined "that would go..." asoka is friendly and redlined "that would go..." which makes perfect sense, FR is asoka's favorite civic. the other three already have state religions.
in my case it's FAR easier to do it that way, with the civic not converting to my religion, since i think i had previous been using a religion that wasn't in a lot of their cities. the major religion on the planet never got to any of my cities. whether convert religion/adopt civic will be easier for you will of course vary, but don't forget that both will end up with the same result.
