If a character is quiet on the outside but loud on the inside, what would be some subtle hints to it

Everybody nowadays seems to think they're "crazy and quirky" because they can't perceive others' similar magnitude of idiosyncrasies. These qualities are but measures of deviation from the norm. Since no precise definition is given of what's unusual anent your character, I've no idea what sort of physical manifestations you could ascribe to them. In general, it could be anywhere from the clothing, hair, manner and subjects of speech & writing, expression of emotions, possessions, habits & hobbies; all the while keeping up with expectations of formality (to some extent) - studies, work, social clique, recreation.
If we're talking live play, I've seen this neat stage technique where onscreen actors would freeze, save for the one character who just monologues their inner world to the audience without the others noticing (or possibly breaking the 4th wall with a minor reference to it now and then). Mayhap dim the lights to emphasise the effect to the viewers.

A lot can be told about the character by what they do when they're alone.