The Botanic Gardens sit at the edge of the Queen's University campus and are free to enter, which makes it easy to underestimate them. The Palm House is the reason to come specifically, a cast-iron and glass curvilinear glasshouse built in the 1840s, one of the oldest of its kind in the world still standing. Inside it's warm and green and completely removed from the city outside.
The wider park is well-kept and popular with students, families, and people walking dogs. On a dry afternoon it fills up in the best way. Bring a picnic if the weather holds, the lawns in front of the Palm House are ideal for it. There's also the Ulster Museum on the park's edge, which is free and worth an hour.
It's the kind of place that's easy to overlook because it asks nothing of you. Show up, wander, sit down somewhere. That's the whole plan.