I can’t seem to come up with much, other than he was nominally an Episcopalian. And:
http://www.geocities.com/peterroberts.geo/Relig-Politics/JMonroe.html
"Summary of Religious Views:
“Franklin Steiner, in his book The Religious Beliefs Of Our Presidents, categorized Monroe among “Presidents Whose Religious Views Are Doubtful.” Indeed, Monroe was very close-mouthed about his religious views – almost nothing can be found in his speeches and writing on that subject.”
Is this really the case?
I visited Ash Lawn, Monroe’s home outside Charlottesville, last year and the tour guide said that Monroe was an Episcopalian, but, as you said, it wasn’t a deeply held faith. Up until Virginia passed the Statute for Religious Freedom, you had to be an Episcopalian to have any real possibility of participating in civic affairs - the first step toward any position of authority was to be named to the vestry of your parish. Even if you weren’t a overly religious person, you participated in parish administration.
I think that Monroe and Washington (who were close) shared the same beliefs - primarily Deists (Washington attended Episcopal services, but never took Communion - he’d get up and leave the service at that point).