Clinics:
http://www.dh.gov.hk/english/tele/tele_chc/tele_chc_shcm.html
take your pick - they're free - but yes you'll have to sit in the waiting room for an hour or so for a walkin consultation. You also have to go back (and wait again) to get the results.
If you want a faster service - Neo Health in Central
http://www.neohealth.com.hk/en/hk-sexual-health-centre.html
but you'll pay $$ for the private-clinic experience. Tests are the same of course, but they'll happily sell you additional tests the government clinic doesn't do, and then you can get the results by email or phone, especially if they're negative.