You cannot say "Кто она верит?" for a very simple reason: Кто is nominative, and hence can only be used to question the subject of a sentence. But then "она" is also nominative, and hence it is the actual subject, making any questions with "кто" meaningless regardless of the verb used. What probably trips you here is the English use of "who", but it might be helpful to remember that in better English the question should actually be "Whom does she believe?".
As for choosing the case (once we have established that it cannot be nominative) -- that is determined by a particular verb, and yes, for "верить" it should be dative.
I believe (no pun intended) that "beliving in somebody/something" is quite different from "believing somebody". I don't believe in Martians, but I can't tell you whether I would believe them until I talk to one. Perhaps one could use "to believe in Jesus Christ" to mean "to entrust oneself to Jesus Christ", but such a use strikes me as highly dated/poetic, something along the lines of "in God we trust".