We have developed and synthesized nanobiocomposite materials based on graphene, poly(3,4-ethylenedioxythiophene), and glucose oxidase immobilized on the surface of various nanomaterials (gold nanoparticles and multi-walled carbon nanotubes) of different sizes (carbon nanotubes of different diameters). Comparative studies of the possible influence of the nanomaterial’s nature on the bioelectrocatalytic characteristics of glucose-oxidizing bioanodes in a neutral phosphate buffer solution demonstrated that the bioelectrocatalytic current densities of nanocomposite-based bioanodes are only weakly dependent on the size of the nanomaterial and are primarily defined by its nature. The developed nanobiocomposites are promising materials for new bioelectronic devices due to the ease in adjusting their capacitive and bioelectrocatalytic characteristics, which allows one to use them for the production of dual-function electrodes: i.e., electrodes which are capable of generating and storing electric power simultaneously.