These two spots make sense to visit together.
As you go down An Duong Vuong St south, just past the HAGL Resort, there's a sharp left turn marked by a gate. Admission to the Queen's Beach area is 5,000 dong, 2,000 more for a motorbike. There's a short climb up a paved road to the top of the headland, where sits the tomb of a famous Vietnamese writer, Han Mac Tu. You can stop for a visit, but it's hardly a highlight, unless you're really into Vietnamese literature. Continuing along, the road leads to the eastern bank of the headland. This is 'Queen's Beach,' named for the wife of Bao Dai, the last king of Vietnam. It's really just a rocky bit of coastline -- we don't know if that's supposed to imply anything about their marriage. There are some cafe's overlooking the water where you can stop for a refreshment, but it's not really a good spot for swimming. You can continue down the road along the headland giving on to the sea for another 2 km -- it makes for a great bike ride or hike -- until you pass through the gate at the bottom of the hill. They may want to see your ticket as you leave, so have it ready. Then you're on Qui Hoa beach, lined with casuarinas trees, and great for swimming. There are some cafes and restaurants along the road, but it's all very peaceful and low key.
The Leprosy Hospital is just off the beach. You can wander around the grounds you like and look at the statues of famous doctors, with a heavy emphasis on Vietnamese and French physicians. There's another hall of notables along the beach, starting with Hypocrites, including the Curies, and a lot of other docs you probably haven't heard of. And just because we know you're wondering, no, there aren't a lot of disfigured people walking around like Night of the Living Dead. At least, we can't be the only ones who were wondering that before we showed up. We hope. It's a state-of-the-art facility where the patients live with their families in little 'chalets.' Visitors are welcome, and those who have taken the tour have found it quite life-affirming.
You can also get to the Leprosy Hospital by not taking the sharp left turn at the end of An Duong Vuong and taking the left fork up the western side of the headland. The entrance to the hospital is well-marked, a hundred metres further down Highway 1A. If you've walked here and don't want to walk back, head to the main gate of the hospital (head west through the hospital if you're on the beach) and ask around there for a ride. It makes for a nice day trip if you're in Qui Nhon for more than the night.