For some class of 2-step Carnot groups $D_n$ with 1-dimensional centre we find the exact values of the constants in $(1,q_2)$-generalized triangle inequality for their $\text{Box}$-quasimetrics ${\rho }_{{\text{Box}}_{D_n}}$. Using this result we get the best version of the Coincidence Points Theorem of $\alpha$-covering and $\beta$-Lipschitz mappings defined on $(D_n,{\rho }_{{\text{Box}}_{D_n}})$.