The electrical conductivity and the electron contribution to the thermal conductivity of metallic hydrogen are calculated over wide ranges of temperatures and densities, including those corresponding to the core of Jupiter. Hydrogen is considered in the liquid and solid states. The applicability of the Wiedemann-Franz law to the description of the metallic hydrogen under these conditions is analyzed. The electron subsystem is described in the framework of the nearly free electron model. The inverse relaxation times for the electrical conduction and the electron contribution to the heat conduction are examined within the perturbation theory to the second order in the electron-proton interaction. The electron-electron Coulomb interaction is taken into account in the random-phase approximation. The exchange interaction and correlations of conduction electrons are included in the local-field approximation. The hard-sphere model is used for describing the proton subsystem. It is revealed that the dynamics of the proton subsystem makes a significant contribution to electron transport phenomena even at temperatures and densities characteristic of the metallic hydrogen under laboratory conditions. This contribution increases with an increase in the hydrogen density.