Here we report a combined study of low-temperature scanning tunneling microscopy and dynamical mean-field theory on PdCrO_{2}, a delafossite metal with an antiferromagnetic order below ∼37.5 K. First, on the CrO_{2}-terminated polar surface we detect a gaplike feature both below and above the Néel temperature. The dynamical mean-field theory calculations indicate that this gap is opened due to the strong correlations of Cr-3d electrons, suggesting the hidden Mott nature of the gap. Then, we observe two kinds of Pd-terminated polar surfaces. One is a well-ordered Pd surface with the Fermi-surface-nesting-induced incommensurate charge modulation, while the other one is a reconstructed Pd surface with the individual nanoscale nonperiodic domain structures. On the well-ordered Pd surface, the interference between the incommensurate charge modulation and the atomic lattice forms the periodic moiré pattern. Our results provide important microscopic information for fully understanding the correlated electronic properties of this class of materials.