  In this paper, we use a geometric technique developed by Gonz\'alez-Prieto,
Logares, Mu\~noz, and Newstead to study the $G$-representation variety of
surface groups $\mathfrak{X}_G(\Sigma_g)$ of arbitrary genus for $G$ being the
group of upper triangular matrices of fixed rank. Explicitly, we compute the
virtual classes in the Grothendieck ring of varieties of the $G$-representation
variety and the moduli space of $G$-representations of surface groups for $G$
being the group of complex upper triangular matrices of rank $2$, $3$, and $4$
via constructing a topological quantum field theory. Furthermore, we show that
in the case of upper triangular matrices the character map from the moduli
space of $G$-representations to the $G$-character variety is not an
isomorphism.
