I don't know if it is something trivial and well-known, or something that depends on more conditions.
Assume $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a multivariate strictly positive function defined in $\mathcal{C}^\infty$ and with a finite integral, i.e. $\int_{\boldsymbol{x}\in \mathbb{R}^d} f(\boldsymbol{x})\text{d}\boldsymbol{x} < K$ for certain $K > 0$.
Consider two functions $g_1$, $g_2$ with the form $$g_i(\boldsymbol{x}) = e^{(\boldsymbol{x}-\boldsymbol{\mu_i})^T\boldsymbol{\Sigma_i}(\boldsymbol{x}-\boldsymbol{\mu_i})},\;\; i\in {1,2}$$ with $\boldsymbol{\mu_i} \in \mathbb{R}^d$ and $\Sigma_i$ a symmetric positive definite $d\times d$ matrix. Can I conclude that if we have $$ \int_{\boldsymbol{x}\in \mathbb{R}^d} f(\boldsymbol{x}) g_1(\boldsymbol{x}) \text{d}\boldsymbol{x} = \int_{\boldsymbol{x}\in \mathbb{R}^d} f(\boldsymbol{x}) g_2(\boldsymbol{x}) \text{d}\boldsymbol{x} $$ then, we must have $g_1(\boldsymbol{x}) = g_2(\boldsymbol{x})$?