This time I have to tell you, you're wrong - sorry.

#!/usr/bin/perl

use strict;
use warnings;

my $str = "A'Bcd";

$_ = $str; s/(')/\Q$1$1/g; print "$_\n";
$_ = $str; s/'/\Q''/g; print "$_\n";
$_ = $str; s/(')/quotemeta "$1$1"/eg; print "$_\n";

__END__
A\'\'Bcd
A\'\'Bcd
A\'\'Bcd
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Abigail