DC Generator on hotel mode ATR 72-600
To start, the FH timer starts when the aircraft takes off (weight-off-wheels) and stops when the aircraft lands (weight-on-wheels).  This means that whether the engine is in hotel mode or running for taxi, the aircraft (and the DC Gen) are not accumulating FHs.  One thing to consider is the difference between an engine hour (EH) and a flight hour.  Engines have hard time limits based on how many hours the engine was run, not on how many hours it is in flight.  One might think that since the DC Gen is directly driven by the engine, that the wear rate should be based on EH, and I might agree with them. However, since the DC Gen is an accessory component (isn't a PWC part, doesn't come with the PW127 engine), the component is lumped in with the rest of the airframe systems, FHs are accumulated just like any other system outside the engine.

You would be right in saying that some aircraft components have hard time limits, that at a certain number of FHs, they need to be replaced. Good examples would be the engine (for overhaul), landing gear and some structure.  The components and the aircraft as a whole lose its certification if the hard time limits are exceeded.  The DC Gen however, more than likely has a soft time limit.  This means that the DC Gen is certified to be on the aircraft until it malfunctions.  The manufacturer of the DC Gen would likely impose a soft time limit, meaning that if the soft time limit is exceeded, it is out of warranty.

When the engine is in hotel mode, the only difference is that the prop and the turbine powering the prop are arrested.  The LP and HP sections of the engine are running as they normally would during taxi.  This post has some helpful images on what the internals of a turboprop look like (reference the second image, the first one is a PT6 I believe).  Hot exhaust air is still rushing past the arrested turbine that would normally power the prop.  This may cause some back pressure that the Full Authority Digital Engine Controller (FADEC) would need to account for, but in terms of the speeds and torque seen by the HP section, accessory drive and DC Gen, the wear seen on the brushes of the DC Gen would be no more than the wear seen during an engine taxi.