We establish the existence of quasi-periodic traveling wave solutions for the $\beta$-plane equation on $\mathbb{T}^2$ with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge, this is the first instance of constructing quasi-periodic solutions for a quasilinear PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.
