You are creating a lot of duplicates. May I offer some humble adjustments:

use strict;
use warnings;

my $MTX = [
       [4,-1,-2,-2, 0,-1],
       [-1,5, 0, 2,-3, 1],
       [-2,0, 6, 1,-3, 5],
       [-2,2, 1, 7,-3, 0],
       [0,-3,-3,-3, 8,-3],
       [-1,1, 5, 0,-3, 9],
      ];

my @result;
my @diagonal = map { $MTX->[$_][$_] } 0..5;

for my $i (0..5) {
  for my $j ($i+1..5) {
    for my $k ($j+1..5) {
      my $sc = 0;
      $sc += $diagonal[$_] for ($i, $j, $k);
      next unless $sc > 17;
      push @result, [$i, $j, $k, $sc];
    }
  }
}

print "@$_\n" for @result;
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