I assume you want to have $f$ to be normalized in some way, otherwise convergence of the maps will not hold even in the case of $L^\infty$ convergence of the complex dilatations. However, even for normalized qc maps, without additional assumptions $L^p$ convergence of the complex dilatations does not imply convergence of the maps.

As an example, consider the family $f_k$ of quasiconformal mappings defined in polar coordinates as $(r,\theta) \mapsto (g_k(r),\theta)$, where $g_k$ is the identity for $r > 1$, $g_k(r) = r^{k^2}$ for $1-\frac1k \le r \le 1$, and $g_k$ is linear on $[0,1-1/k]$. Then $f_k$ is quasiconformal in the plane, $\mu_k$ is supported on the annulus $1-\frac1k \le r \le 1$, so $\mu_k \to 0$ in $L^p$ for any $p<\infty$, and $f_k$ converges to the identity outside the unit disk, and to the constant $0$ inside the disk, so the limit is not even continuous.

Maybe you want to require that the $f_k$ are uniformly quasiconformal, i.e., that there exists $\kappa<1$ such that $\| \mu_k \|_\infty \le \kappa$ for all $k$?