use Encode qw( encode_utf8 );

my $encoded = encode_qp(encode_utf8($str));

##</code><code>##

$ cat a.pl
use strict;
use warnings;
use feature qw( say );

use utf8;
use open ':std', ':encoding(UTF-8)';

use Encode            qw( encode_utf8 );
use MIME::QuotedPrint qw( encode_qp );

for my $str (
   "a\x{201E}z",
   "a„z",
   "a\x{201C}z",
   "a\x{03C9}z",
   "aäz",
) {
   say $str;

   my $str_q;
   if (!eval {
      $str_q = encode_qp($ARGV[0] ? encode_utf8($str) : $str);
      return 1;
   }) {
      warn($@);
      next;
   }

   print $str_q;
}