I might be wrong, but I am under the impression that you're not considering that queens can attack in columns as well as rows. You should also consider attacking squares that are "before" your current position, like in this untested snippet:

sub mark_attacks {
   my ($r,$c,$array) = @_;

   $array->[$r]->[$c] = 'Q';

   # mark everything

   my $none;

   for my $i (1 .. $SIZE) {
      $none = 1;
      $none = $array->[$r]->[$c + $i] = 0 if ($c + $i < $SIZE);
      $none = $array->[$r]->[$c - $i] = 0 if ($c - $i >= 0);
      $none = $array->[$r + $i]->[$c] = 0 if ($r + $i < $SIZE);
      $none = $array->[$r - $i]->[$c] = 0 if ($r - $i >= 0);
      $none = $array->[$r - $i]->[$c - $i] = 0 if ($r - $i >= 0 and $c
+ - $i >= 0);
      $none = $array->[$r + $i]->[$c + $i] = 0 if ($r + $i < $SIZE and
+ $c + $i < $SIZE);
      $none = $array->[$r - $i]->[$c + $i] = 0 if ($r - $i >= 0 and $c
+ + $i < $SIZE);
      $none = $array->[$r + $i]->[$c - $i] = 0 if ($r + $i < $SIZE and
+ $c - $i >= 0);
      last  if $none;
   }
}
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I also think than a single loop might improve performance slightly.

Good luck in your assignment.