The three-dimensional nonlinear size-dependent motion characteristics of a microbeam are investigated numerically, with special consideration to one-to-one internal resonances between the in-plane and out-of-plane transverse modes. All of the in-plane and out-of-plane displacements and inertia are taken into account and Hamilton's principle, in conjunction with the modified couple stress theory, is employed to obtain the nonlinear partial differential equations governing the motions of the system in the in-plane and out-of-plane directions. The discretization procedure is carried out by applying the Galerkin technique to the partial differential equations of motion, yielding a set of nonlinear ordinary differential equations. A linear analysis is performed upon this set of equations so as to obtain the size-dependent natural frequencies of the system. The nonlinear analysis of the discretized equations of motion is carried out by employing the pseudo-arclength continuation technique, resulting in the resonant responses of the system. It is shown that, due to the presence of one-to-one internal resonances between the in-plane and out-of-plane transverse modes, an in-plane excitation can give rise to an out-of-plane displacement; the internal resonances also cause the occurrence of extra solution branches and new bifurcation points.