In this study, we report the synthesis of two new copper complexes: [Cu(C₁₁H₇O₂)(SCN)(C₁₀H₈N₂)], denoted as (C-1), and [Cu(C₁₁H₇O₂) (C₁₂H₈N₂) Cl]·H₂O, denoted as (C-2). They are based on 2,2′-bipyridine or 1,10-phenanthroline and 2-hydroxy-1-naphtaldehyde ligands. The obtained complexes were characterized by FT-IR, UV-visible spectroscopy, and single-crystal X-ray diffraction analysis. Molecular docking was employed to predict the binding mode involved in the interaction between the two synthetic copper (II) complexes and HIV-1 protease enzyme. The X-ray structural analysis revealed that the crystal structures of both complexes are mainly stabilized by several intra- and intermolecular hydrogen bonds. The fingerprint plots associated with the Hirshfeld surfaces of both complexes clearly show that H···H interactions provide the largest contributions. According to the docking results, the synthesized complexes exhibit promising features which enable them to be bound to the HIV-protease enzyme.