Extensions of Matrix-Tree Theorem
Let me give a side answer to your question. Consider the variant where you only care about edge-minimal spanning trees: for 2 terminals $s$ and $t$, the problem is equivalent to counting simple $s,t$-paths, which is $\# P$-complete; also, the parameterized version where you ask for paths of length $k$ is $\# W[1]$-hard. If you care about arbitrary $T$-steiner trees (i.e. not necessarily minimal), I expect that it is also $\# P$-hard though I'm not aware of any reference.
These hardness results suggest that it's unlikely to find some "nice formula" for the number of $T$-steiner trees.

The references are: "The complexity of enumeration and reliability problems" by L. G. Valiant, and "The parameterized complexity of counting problems" by J. Flum and M. Grohe.