We presented a novel $\mathcal{N}=2$ ${\mathbb{Z}}_2^2$-graded supersymmetric quantum mechanics ( ${\mathbb{Z}}_2^2$-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $\mathcal{N}=2$ ${\mathbb{Z}}_2^2$-supersymmetry algebra. The ${\mathbb{Z}}_2^2$-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional ${\mathbb{Z}}_2^2$-supersymmetric Lagrangian of $\mathcal{N}=1$, which we constructed in our previous work. The ground states of the ${\mathbb{Z}}_2^2$-SQM were also investigated.