Does Hardened Scales trigger when +1/+1 counter are moved from one permanent to another?
Yes, Hardened Scales triggers off moving +1/+1 counters. From the official rules:


  121.5. If an effect says to “move” a counter, it means to take that counter from the object it’s currently on and put it onto a second object. If the first and second objects are the same object, nothing happens. If the first object has no counters, nothing happens; the second object doesn’t get a counter put on it. If the second object (or any possible second objects) is no longer in the correct zone when the effect would move the counter, nothing happens; a counter isn’t removed from the first object.


Note that Hardened Scales only triggers once per batch of counters moved, so if several counters are moved at once, it will not double the amount, but only add 1.