Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70’s by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field ξ = (ξ(x))x∈Zd of i.i.d. random variables, which is called the random scenery, and a random walk S = (Sn)n∈N evolving in Z , independent of the scenery. The RWRS Z = (Zn)n∈N is then defined as the accumulated scenery along the trajectory of the random walk, i.e., Zn := ∑ n k=1 ξ(Sk). The law of Z under the joint law of ξ and S is called “annealed”, and the conditional law given ξ is called “quenched”. Recently, functional central limit theorems under the quenched law were proved for Z by the first two authors for a class of transient random walks including walks with finite variance in dimension d ≥ 3. In this paper we extend their results to dimension d = 2.