  This paper investigates the relations between the Toda conformal field
theories, quantum group theory and the quantisation of moduli spaces of flat
connections. We use the free field representation of the $\mathcal{W}$-algebras
to define natural bases for spaces of conformal blocks of the Toda conformal
field theory associated to the Lie algebra ${\mathfrak s}{\mathfrak l}_3$ on
the three-punctured sphere with representations of generic type associated to
the three punctures. The operator-valued monodromies of degenerate fields can
be used to describe the quantisation of the moduli spaces of flat
$\mathrm{SL}(3)$-connections. It is shown that the matrix elements of the
monodromies can be expressed as Laurent polynomials of more elementary
operators which have a simple definition in the free field representation.
These operators are identified as quantised counterparts of natural higher rank
analogs of the Fenchel-Nielsen coordinates from Teichm\"uller theory. Possible
applications to the study of the non-Lagrangian SUSY field theories are briefly
outlined.
