no.
page 20 of rulebook
"After fully resolving the Hero’s attack, the Monster may moveup to its full Movement rating (as listed on its card) and then
attempt an attack against the Hero who attacked him (it cannot
move after attacking). A moving Monster may choose to
open doors and use portals or move in any direction he wishes,
following the same Movement rules and restrictions as Heroes.
The player controlling the Monster doesn’t have to move it or
even perform its attack, if he so desires, but any attack made
by the Monster must necessarily have the active Hero as the
primary target (see example on page 22)."