Yes, I think people can say things like "wow!" or "good grief!" or other things in either a quiet voice, or under their breath—or even to themselves. You could portray this in lots of ways, from 'quietly exclaim,' as you said in your title, to actually illustrating it: "What did you just say?" "I said 'wow.'"
I would support your usage of 'quietly exclaim'. It sounds like an oxymoron, but it is really not. 'Quietly shouted' would be a problem, although, I don't imagine too many people would have an issue with 'whispered loudly'. Personally, I like the unexpected juxtaposition of your phrase.