abowley@lave:~$ perl -e '$a = $b = "word?dude"; print "$a $b\n" if $a 
+=~ m/\Q$b/'
word?dude word?dude
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That's not necessarily a bad way to do it, but everyone should be aware that if there are any metacharacters in $b and \E precedes them, the metacharacters will be parsed as metacharcters.
IOW, if $b == "???\E?", the fourth '?' will be not be matched literally.

The only real advantage to punk music is that nobody can whistle it.

I was skeptical about this warning, and a simple experiment shows that it is not accurate:

  DB<1> $x = '???\E???'
 
  DB<2> p "\Q$x"
\?\?\?\\E\?\?\?
  DB<3>
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A \Q cannot be ended by a \E in an interpolated string. And, using B::Deparse's -q option in 5.6, we can see why:

~> perl -MO=Deparse,-q -e '$x = q{?\E?}; $y = "a\Q$x\Eb"; print "$y\n"
+'
$x = '?\\E?';
$y = quotemeta $x;
print $y . "\n";
-e syntax OK
[download]

This shows that \Q\E inside a double-quoted string is actually compiled as concatenation and quotemeta. The value of $x at runtime won't change the scope of the quotemeta().