Do you mean the hotel limo service for a fee or do you mean courtesy vehicles for short complimentary trips?

AFAIK the Peninsula HK vehicles are limos for which you must pay, although there could be some packages that include airport transportation or you might get complimentary rides as a benefit of booking certain suites.

In LA, other luxury hotels such as Raffles Hermitage do offer local free limo service for very short trips, so it wouldn't surprise me if the Beverly Wlitshire does the same, probably within about three miles of the hotel and certainly not including the airport..