<p>A hyperbolic framed curve in hyperbolic 3-space is defined as a smooth curve equipped with a moving frame. Such curves may possess singularities. Leveraging this moving frame, we can delve into the differential geometric properties of hyperbolic framed curves, even at their singular points. Specifically, we utilize Legendrian dualities to examine the focal surfaces and evolutes of hyperbolic framed curves. The focal surfaces are the dual counterparts of the tangent indicatrix of the original hyperbolic framed curves. Additionally, we introduce the concept of the Legendrian dual surfaces of the evolutes of hyperbolic framed curves. Furthermore, we present classifications of the singularities of several dual surfaces. We also establish the relationships among the focal surfaces, evolutes, and the dual surfaces of the evolutes. Finally, we conduct an in-depth exploration of the duality relations of singularities between the focal surfaces and the dual surfaces of the evolutes.</p>