We investigate an a posteriori error analysis of adaptive finite element approximations of linear-quadratic boundary optimal control problems under bilateral box constraints, which act on a Neumann boundary control. We use a symmetric interior Galerkin method as discretization technique. An efficient and reliable residual-type error estimator is introduced by invoking data oscillations. We then derive local upper and lower a posteriori error estimates for the boundary control problem. Adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical results are presented to illustrate the performance of the adaptive finite element approximation.