  Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity
formula is an alternating sum over the Weyl group whose terms involve the
$q$-analog of Kostant's partition function. For $\xi$ (a weight of
$\mathfrak{g}$), the $q$-analog of Kostant's partition function is a
polynomial-valued function defined by $\wp_q(\xi)=\sum c_i q^i$ where $c_i$ is
the number of ways $\xi$ can be written as a sum of $i$ positive roots of
$\mathfrak{g}$. In this way, the evaluation of Kostant's weight
$q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a
highest weight representation of $\mathfrak{g}$. In this paper, we give closed
formulas for computing weight $q$-multiplicities in a highest weight
representation of the exceptional Lie algebra $\mathfrak{g}_2$.
