We report on a comprehensive thermodynamic study of a quasi-two-dimensional (quasi-2D) quantum magnet Cu2(OH)3Br which in the 2D layer can be viewed as strongly coupled alternating antiferromagnetic and ferromagnetic chains. In an applied magnetic field transverse to the ordered spins below TN=9.3 K, a field-induced phase transition from the 3D ordered to a disordered phase occurs at Bc=16.3 T for the lowest temperature, which is featured by an onset of a one-half plateaulike magnetization. By performing quantum Monte Carlo simulations of the relevant 2D model, we find that the plateaulike magnetization corresponds to a partial symmetry restoration and the full polarization in the ferromagnetic chains. Our numerical simulations also show that the magnetization saturation occurs with full symmetry restoration at a much higher field of Bs≃95 T, corresponding to a 1D quantum phase transition in the antiferromagnetic chains. We argue that the experimentally observed field-induced phase transition at Bc follows from the partial symmetry restoration and the concomitant dimensional reduction.