As a native Japanese speaker, I'd rather explain my own feeling or thoughts at the times facing of someone saying those sentences:
He/she has "made" him/herself believe something - or he/she consiously defines his/her beliefs as such.
2) いつか願いが叶うのを信じている。/ いつか願いが叶うことを信じている。
He/she believes something.
Basically those sentences are fairly similar but I slightly sense of more consciousness on #1 than #2. Perhaps "と" makes me feel of psychological distance between subjects and objects, thus revealing existense of implied forces (i.e., will) tighten them up and turning objects into subjects (or "heartfelt voice"). On the other hand, objects on #2 are still objects.
If the verb "信じている" is changed to some other, for example "考えている", I think the distinction between two becomes clearer. I'm not sure this can be applied to "信じている" but could be understood as some inherent meanings of "信じる" has masked out grammatical differences - probably "信じる" is subjective after all.