I recommend Kimhae Choi Eye Clinic.

There's probably no one in Gimhae who doesn't know Choi An. I've been going there since I was very young. The teachers here are friendly and skilled. I once had an eye exam, and they gave me a thorough, hour-long examination. Even when I was in school, I would wear my gym uniform during lunchtime and visit the clinic. After my grandmother had cataract surgery, they even picked up every single strand of hair on the floor. I highly recommend them.