Oreo, to see a doctor you need to just pick your doctor and make an appointment. Some are very busy and may give you an appointment the next week, others might see you the next day. There are lots of threads on this forum about recommended doctors.
To register at Queen Mary's you need to be residing in that catchment area. I think you can figure out which catchment area from the Hospital Authority website. You can call them (the number should be on the HA website) and tell them when you are due and make an appointment. If you fall in the catchment area - you might be asked to provide proof of residence - they are obliged to take you in. They will then schedule an appointment for you. I think they require a confirmation from a doctor (could be a general practitioner) that you are pregnant.