The functionality outlined below, and much more, is implemented in Hodge diamond cutter, which can be used in Sage. If you use it for your research, please cite it using DOI.

Last week Massimo Bagnarol, Barbara Fantechi, Fabio Perroni: On the motive of zero-dimensional Quot schemes on a curve appeared on the arXiv, and they study moduli spaces of sheaves on a curve which I hadn't seen before (at least not for their intrinsic geometry, only as a tool). They describe the class in the Grothendieck ring of $\mathrm{Quot}_C^n(\mathcal{E})$, where $C$ is a smooth projective curve of genus $g$, $\mathcal{E}$ is a vector bundle of rank $r$ on $C$ and $n\geq 1$ is the length of the finite-dimensional quotients of $\mathcal{E}$ which are parametrised.

They show that the class in the Grothendieck ring is independent of $\mathcal{E}$, effectively reducing the computation to an earlier result of Bifet for $\mathcal{E}=\mathcal{O}_C^{\oplus r}$. The formula (see proposition 4.5) is \begin{equation} [\mathrm{Quot}_C^n(\mathcal{E})] = \sum_{\mathbf{n}\in\mathbb{N}^r,|\mathbf{n}|=n} [\operatorname{Sym}^{n_1}C]\cdots[\operatorname{Sym}^{n_r}C]\mathbb{L}^{d_{\mathbf{n}}} \end{equation} where $d_{\mathbf{n}}:=\sum_{i=1}^r(i-1)n_i$, so one can then apply the motivic measure $\mathrm{K}_0(\mathrm{Var}/k)\to\mathbb{Z}[x,y]$ given by the Hodge polynomial to get the Hodge diamond. For all of you who wish to use it in Hodge diamond cutter I have good news, as I just added it.

Let me know about any requests you might have!