Derivative of Tutte polynomial at -1

This polynomial isn't the usual Tutte polynomial, but it's equivalent. Provided that $G$ is connected (which is probably the case you're interested in, and I'll assume), it looks like $f_G(q,v)=qv^{n-1}T_G(q/v+1,v+1)$, where $n$ is the number of vertices of $G$.

One thing that comes to mind is the Crapo beta invariant, which is $(-1)^n\chi_G'(1)$, where $\chi_G(k)=(-1)^{n-1}kT_G(1-k,0)$ is the chromatic polynomial. (E.g., see exercise 22 of lecture 4 of Stanley's notes on hyperplane arrangements: http://www-math.mit.edu/~rstan/arrangements/arr.html.) Your polynomials, especially $g$, might be related to this, but I haven't thought about the details.