Nonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.