I guess this is it for my MO lurking. So anyway, you're interested in seeing example calculations, similar to Turaev IV.5.4, of the action introduced in Turaev IV.5.1, right?

This action is also referred to known as the quantum representation of the mapping class group and has been considered from numerous viewpoints including of course TQFTs via quantum groups, the skein theory of the Kauffman bracket, conformal field theory, and geometric quantization. Calculations in the higher genus case grow messy quickly, but the skein theory approach provides an algorithm (cf. Masbaum-Vogel, *3-valent graphs and the Kauffman bracket*) for calculating explicitly the representation in a particular natural basis of the vector spaces associated to the boundary surface. This algorithm has been implemented by A'Campo and Masbaum and should be available here -- the site seems to be down at the moment, but leave me an e-mail if you want a copy of the (usually freely available) program. How it works and some example calculations have been explained by A'Campo.

Note that all calculations of this program are performed using the skein theory approach, as described in Turaev Ch. XII. Off the top of my head, the only explicit non-torus and non-computer assisted calculations of the quantum representations I remember seeing are for the four-holed sphere (cf. Masbaum, *An element of infinite order in TQFT-representations of mapping class groups*, Andersen-Masbaum-Ueno, *Topological quantum field theory and the Nielsen-Thurston classification of $M(0,4)$*, Laszlo, Pauly, Sorger - *On the monodromy of the Hitchin connection*). By the factorization properties of the TQFT/the quantum representations, this then gives example calculations for all surfaces of genus $g \geq 2$, as the four-holed sphere embeds in such.

Now, a different family of examples is provided by complements of links in $S^3$, i.e. 3-manifolds having boundary a disjoint union of tori. Here, in the modular category corresponding to, say $U_q(\mathfrak{sl}_N)$ or the one arising from the HOMFLY polynomial (cf. Blanchet - *Hecke algebras, modular categories and 3-manifold quantum invariants*), knowing the vector in the vector space associated to (a disjoint union of) tori corresponding to the knot complement more or less boils down to understanding the coloured HOMFLY invariants of the link in question, as the vector space of the torus has a basis of handlebodies containing longitudinal (coloured) links, and these link invariants have been considered by several people (in particular in the case $N = 2$).