Not necessarily. It won't work for

print($regexp)  Expect        Gives
--------------  ------------  ----------
\Q*\E\Q*\E      \*\*          \*\Q*\E
\Q**LOL**       \*\*LOL\*\*   \Q**LOL**
\\Quit\\Exit    \\Quit\\Exit  \Quit\Exit
\Qfoo\\E        foo\\\\E      foo\\
[download]

Solution:

my $in_quote = 0;

$regexp =~ s/([^\\]|\\.)/
   if ($in_quote) {
      if ($1 eq '\\E') {
         $in_quote = 0;
         ''
      } else {
         quotemeta($1)
      }
   } else {
      if ($1 eq '\\Q') {
         $in_quote = 1;
         ''
      } else {
         $1
      }
   }
/eg;
[download]

Update: Alternative:

$regexp =~ s/
   \G
   (
      (?:[^\\]|\\[^Q])*
   )
   (?:
      \\Q
      (?:[^\\]|\\[^E])*
      (?:\\E)?
   )?
/
   $1 . (defined($2) ? quotemeta($2) : '')
/xge;
[download]

Neither snippet is fully tested. In fact, both are known to be unable to handle regexps in which (?{...}) or (?{{...}}) are used. What's wrong with what I suggested in my earlier post?