This all depends on where the pet is going or coming from.
If coming to Hong Kong from a group 3 country, then there is mandatory 120 days quarantine at one of 2 government quarantine facilities. Please also be aware there is usually a 12-18 month wait for a quarantine space (longer for dogs) and there is no exemption. Please note that mainland China and Macau are amongst those group 3 countries. The list of group 1,2 & 3 countries can be found on the Hong Kong AFCD website.
If leaving from Hong Kong to other countries, then please check with the arrival countries requirements. UK, USA, EU, Japan, Singapore don’t require quarantine but some countries such as Australia, Taiwan, Malaysia, New Zealand amongst a few others do require a short quarantine stay of 7-10 days.