In the article, we establish some identities involving special values of multiple zeta functions among the counting functions of number of representations of an integer by a linear combination of figurate numbers such as triangular numbers, square numbers, pentagonal numbers, etc. More precisely, we provide our result for $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ (for a fixed $a \ge 3$), the number of representations of $n$ as a sum of $k$-triangular numbers, as a sum of $k$-square numbers and as a sum of $k$-higher figurate numbers (for a fixed $a \ge 3$), respectively. Moreover, these identities also occur when one of $\delta_k(n)$, $r_{k}(n)$ and $\mathcal{N}_{k}^{a}(n)$ is replaced by the $k$-colored partition functions.