Energy localization due to nonlinearity of atomic interactions in a graphene sheet is investigated. In nonlinear lattices, localized vibrational modes called discrete breather (DB) or intrinsic localized mode (ILM) exist. We search the DB in a graphene sheet as aperiodic solution with larger frequency than phonon modes by a procedure combining the Newton-Raphson method and molecular dynamics (MD) method. We obtain long-alive vibrational modes. Some characteristics of the localized mode such as spatial structure, amplitude and vibrational frequency are discussed. Adding to this, we investigate the linear stability of the DB solutions by numerical analysis based on Floquet theory of periodic solutions. Dynamical instability of the vibrational mode is observed by MD simulations, which may lead to change of the structure of DB.