In this paper, we perform a theoretical investigation of the plasmon coupling in silver (Ag) nanocube–nanosphere dimer using finite difference time domain (FDTD) technique. Due to plasmonic Fano resonance, we observe a pronounced dip in the absorption spectrum, which is induced by the destructive interference between the bright dipole mode from the Ag nanocube and the dark quadrupole mode from the Ag nanosphere. We study the effects on the Fano resonance by varying the dimensions of the nanocube and nanosphere, as well as the inter-surface gap distance between them. In addition, we vary the local dielectric environment surrounding the dimer and find a wide tuning in the Fano resonance line-shape. We calculate the localised surface plasmon resonance (LSPR) sensitivity of the nanocube–nanosphere dimer to the surrounding environment and find a high figure of merit (FOM) of 32.23, which indicates its promising potential as a plasmonic biosensor.