The word ”conversion” does not always refer to the changing of the formal membership of a religion, but the change from a “worldly” to a religious life. This may be used more often among evangelical protestants, but catholics sometimes does it to; I have seen biographies of StFrancis and Ignatius of Loyola refer to their religious awakenings as “conversion”.
“Democracy” may mean a lot of things. If it means that humanity is the source of ultimate authority, it is irreconcilable with the abrahamite religions (and also hinduism, cofusianism etc). If it just means that free elections is the best way to appoint public officials of various kinds there is no problem at all. If there is something in between, it may vary with religion and individual.