Computing such induced norms is a hard problem. For the case of $p=2$ and $q \ge 2$, have a look at [this paper by Barak et al.][1] to see how tricky the problem is.

Typically, for other than the nice cases of $1,2, \infty$ style, these norms are NP-hard to compute, with well-known results for the case $p \ge q$ (see also e.g.: ["Matrix norms are NP-Hard to approximate"][2]). The paper of Barak et al focuses on the *hypercontractive* case of $p < q$.

For the specific case of $p=2$ and $q=\infty$ mentioned above, <s>I think</s> the paper of Barak et al cited above mentions that the norm is just the largest 2-norm of any row of the matrix (thanks to N. Johnston for pointing this out).


  [1]: http://arxiv.org/abs/1205.4484
  [2]: http://dspace.mit.edu/openaccess-disseminate/1721.1/64956