<p>Topological (Thouless) pumping of solitons is a unique phenomenon that enables quantized, controllable transport of localized excitations in underlying dynamical potentials. While this effect has been extensively studied in Kerr media, it remains unexplored in <i>χ</i>⁽²⁾ nonlinear materials with parametrically interacting fields. Here, we predict the pumping of two-dimensional quadratic solitons induced by mutually sliding shallow lattices. We identify three distinct pumping regimes: the absence of transport for small-amplitude solitons, non-quantized transport in a transient regime at intermediate amplitudes, and robust quantized transport for solitons with sufficiently large amplitudes. The quantized transport regime depends on the phase mismatch between the fundamental and second-harmonic waves. Pumping along orthogonal directions with periods whose ratio approximates an irrational number leads to quantized transport whose direction converges to that defined by the corresponding irrational ratio, effectively realizing transport associated with truly incommensurate longitudinal periods. These results open new prospects for observing topological transport of parametrically interacting fields in arbitrary directions.</p>