Algorithmically, you can think of this as a tensor product of 2 2x2 matrices. In code, you could represent this as

for my $a (0,1) {
   for my $b (0,1) {
      print "Quadrant $a,$b:\n";
      for my $i (0,1) {
         for my $j (0,1) {
            my $row = 2*$a + $i;
            my $col = 2*$b + $j;
            print "  element $i,$j => matrix coord $row, $col\n";
         }
      }
   }
}
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If you wanted to convert in the opposite direction, just use the inverse equations:

for my $row (0..3) {
   for my $col (0..3) {
      print "Quadrant ", int($row/2),',',int($col/2);
      print "  Element ", $row % 2,',', $col % 2,"\n";
   }
}
[download]

Update: added inverse operation.