No, \Q...\E doesn't escape $ and @. You can use quotemeta instead:

like('asdf$', quotemeta('asdf$'));
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If you absolutely need to compile the regexp before calling like (doubtful), here are some possible calling syntaxes:

$re = quotemeta('asdf$'); like('asdf$', qr/$re/);

   -or-

like('asdf$', qr/@{[ quotemeta('asdf$') ]}/);

   -or-

like('asdf$', map { qr/$_/ } map quotemeta, 'asdf$');
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