What I'm trying to do, is select column [ 13] with missing information and column [ 14] with no missing information

Unfortunately your line which does this is missing a digit. You have if ($column[13] eq "" && $column[4] ne "" ) which would do this if it said [14] instead of [4]. Is this your first problem?

#!/usr/bin/perl
use warnings;
use strict;

my $filename = 'pedigree_proband_testfile.txt';

open (FILE, "<", $filename) or die "Cannot open file $!";

my @data = <FILE>;
my @column;
my $motherID;

foreach my $line (@data)
{
    @column = split ( /\t/, $line);

    if ($column[13] eq "" && $column[14] ne "" )
    {
        print "DEBUG: $line";

        # It isn't clear what you are trying to do below here
        $motherID = join($column[2],$column[4]);

        if ("$motherID" eq "$column[2]")
        {
            print "$line\t\n";
        }
    }
}
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See item number 2 on the Basic debugging checklist for the use of debugging aids like this print statement.