Hi there - congratulations and welcome to Hong Kong! I was in a very similar situation 4 months ago, I got a positive pregnancy test 5 days after I arrived. Because I had some bleeding and needed to see someone urgently, I just phoned the Matilda hospital and saw the first doctor available. This happened to be Dr Patrick Chan who I was very happy with. If you search this website you will find recommendations of other doctors too. The doctor you choose will only be able to deliver at certain private hospitals and he or she can tell you what these are. The next step for me was to call the hospitals and arrange to take a look around to see what suits your needs. My favourite of the private hospitals was the Matilda, purely because the baby can stay with you the whole time you stay there. A lot of the other private hospitals keep the baby in a nursery apart from feeding times, this didn't suit me - but everyone's different.
I found the idea of giving birth Hong Kong a pretty scary prospect at first, being newly arrived from the UK. The Matilda is lovely and the idea of going there made me feel very comfortable. However, as we got more used to being in Hong Kong I changed my mind and am now using the public system and going to the Queen Mary hospital. So, although you don't get the same level of comfort in the public system, the medical care you receive is as good if not better than many private hospitals can provide. I feel very confident that I will have a good experience at the Queen Mary. (I should add that we don't have maternity insurance so would have had to pay the full cost of a private hospital which can be expensive and pretty unpredictable - I'm glad we decided to save the money!).
I'm sure others will reply with recommendations on doctors you might like to use, I've heard good things about Dr Joe Chan and Dr Sally Ferguson as well as Patrick Chan who I used. Once you get the ball rolling with a doctor's appointment everything will start to fall into place. Good luck.