encode_qp expects encoded text ("the quoted-printable encoding is only defined for single-byte characters").

use Encode qw( encode_utf8 );

my $encoded = encode_qp(encode_utf8($str));
[download]

---

$ 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;
}
[download]

$ perl a.pl 0
a„z
Wide character in subroutine entry at a.pl line 22.
a„z
Wide character in subroutine entry at a.pl line 22.
a“z
Wide character in subroutine entry at a.pl line 22.
aωz
Wide character in subroutine entry at a.pl line 22.
aäz
a=E4z=

$ perl a.pl 1
a„z
a=E2=80=9Ez=
a„z
a=E2=80=9Ez=
a“z
a=E2=80=9Cz=
aωz
a=CF=89z=
aäz
a=C3=A4z=

---

Note that "aäz" didn't actually work; Perl was just unable to automatically detect the error.