Try something like this (semi-tested):

use warnings;
use strict;

my $datamat =
  "He said, »Someone once said ›This is what they said‹ but I say some
+thing else.«";
print qq{'$datamat' \n\n};

my $i = 0;

$datamat =~ s{ » (.*?) « }
             { ++$i;  qq{<q marker="»" sID="$i"/>$1<q marker="«" eID="
+$i"/>}; }xmsge;

$datamat =~ s{ › (.*?) ‹ }
             { ++$i;  qq{<q marker="›" sID="$i"/>$1<q marker="‹" eID="
+$i"/>}; }xmsge;

print qq{[[$datamat]] \n\n};
[download]

(Assumes entire file has been read (i.e., "slurped" (update: see also File::Slurp)) into the  $datamat variable.)
(Update: Also assumes matching  » ... « and  › ... ‹ quote character pairs are never nested!)

Update: Please see perlre, perlretut, and perlrequick.

Update 2: A regex expression like  » (.*?) « may run faster if written as  » ([^«]*) « instead.