We studied the transfer of oxygen vacancy and proton in Y-doped BaZr$O_3$ (BYZ) using density functional theory (DFT). An oxygen vacancy was generated in the $2{ imes}2{ imes}2$ BYZ superstructure by replacing two Zr atoms with two Y atoms to satisfy the charge neutrality condition. The O vacancy transfer between the first and second nearest O atom sites from a Y atom showed the lowest activation energy barrier of 0.42 eV, compared to the other transfers between first and first, and second and second in the superstructure. Two protons were inserted in the structure by adding a proton and hydroxyl that were supplied by the dissociation of a water molecule. The two protons bonded to the first and second nearest O atoms were energetically the most favorable. The activation energy barrier for a proton transfer in the structure was 0.51 eV, when either proton transferred to its neighbor O atom. This value was well matched with the experimentally determined one.