How to calculate he equilibrium constant change for acid-base reactions in a different solvent?
For all solvents that can dissociate into ions, a dissociation constant can indeed be measured. Expect them to be unpredictable in each direction.

Furthermore, the $\mathrm pK_\mathrm a$ values you know are typically those measured in water, although some organic chemists rely more heavily on the DMSO tables e.g. the one by Evans. Just a quick look at the first column shows you that there is no way you can predict the $\mathrm pK_\mathrm a$ value in one solvent from the one in a different solvent:

$$\begin{array}{lcc}\hline
\text{compound} &amp; \mathrm pK_\mathrm a(\ce{H2O}) &amp; \mathrm pK_\mathrm a(\ce{DMSO}) \\ \hline
\ce{HF} &amp; 3.17 &amp; 15\phantom{.0}\\
\ce{HCN} &amp; 9.4\phantom{0} &amp; 12.9\\ \hline\end{array}$$

While $\ce{HF}$ is the stronger acid in water, $\ce{HCN}$ is the stronger acid in $\ce{DMSO}$.

Couple this with a different dissociation constant means that you can calculate nothing ab initio.