FWIW, single-quoted regex notation gets you closer (update: than double-quoting) to the desired final form, but still not all the way and there are some pitfalls. Better, IMHO, to stick to qr//. (BTW: morgon: It's not necessary to escape a  " (double-quote) in a qr// expression.)

File rx_qr_qq_q_1.pl:

use warnings;
use strict;

my $rx_qr = qr/(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\")))/;
my $rx_qq = qq/(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\")))/;
my $rx__q =  q/(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\")))/;

print qq{raw qr: $rx_qr \n};
print qq{qr/qq/: }, qr/$rx_qq/, "\n";
print qq{qr/q/ : }, qr/$rx__q/, "\n";
[download]

Output:

c:\@Work\Perl\monks\morgon>perl rx_qr_qq_q_1.pl
raw qr: (?-xism:(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\"))))
qr/qq/: (?-xism:(?:(?|(?:")([^\"]*(?:\.[^\"]*)*)(?:"))))
qr/q/ : (?-xism:(?:(?|(?:\")([^\\"]*(?:\.[^\\"]*)*)(?:\"))))
[download]