We theoretically propose two possible routes to realizing quantum anomalous Hall states in altermagnetic materials. We consider a minimal square-lattice Hubbard model with antisymmetric spin-orbit coupling associated with an orthorhombic crystal structure, which supports a topologically trivial altermagnetic state. By incorporating Rashba-type spin-orbit coupling and external perturbations, we demonstrate that this trivial state can be turned into topological altermagnetic phases in two distinct ways. The first route is driven by a staggered potential that breaks the symmetry connecting crystallographically equivalent sublattices, leading to a topological altermagnetic ground state characterized by a quantized Hall conductivity | σ_xy |=e²/h and a Chern number C=1. The second route is realized by applying a magnetic field perpendicular to the two-dimensional plane. The resulting topological state appears as a metastable state in the magnetic hysteresis loop, exhibiting a quantized Hall conductivity | σ_xy |=2e²/h associated with a Chern number C=2. We show that these topological transitions are accompanied by characteristic gap closings at the Brillouin-zone boundary, with the number of gap-closing points determining the Chern number. Ribbon-geometry calculations reveal chiral edge states consistent with the bulk topological invariants and demonstrate distinct spin polarizations between the C=1 and C=2 states. Our results establish experimentally accessible routes to quantized anomalous Hall responses in altermagnets.