Is exposed doorbell wiring in the garage dangerous?
Can you get electrocuted from exposed wires?

Yes, absolutely, if the wiring is powerful enough.  It's not necessarily very LIKELY that doorbell wiring is that powerful but it's possible, depending on how stupid the installer was.

As for putting a box over them - in the US exposed permanent wiring is a flat out no no.  

But - based on your question - I'm going to assume that you have minimal to no experience with electricity, so I URGE you, do not try to DIY.  Hire a professional electrician.