join($column[2],$column[4])
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join EXPR,LIST

    Joins the separate strings of LIST into a single string with fields separated by the value of EXPR, and returns that new string

You probably wanted:

$motherID = "$column[2]$column[4]" # Interpolation
# OR (They both do the same thing)
$motherID = $column[2] . $column[4]; # Concatenation
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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.

So, as a clarification step, I wrote code to print the columns of interest (the ones that you mentioned).
In the code below, I translated the column indices into names based upon your post. Usually it is a good idea to use a meaningful name for a column instead of a number just because that is much easier for humans to understand. Perl is very flexible in how this can be done.

Please explain what the desired output is for each line below.

Also, in my opinion a tab delimited file is a difficult critter to deal with because it is difficult to see the difference between space and tab. I guess you are exporting this from an Excel table? Better is to use standard CSV or I often use pipe "|" delimiters for files that may be subsequently modified by humans. Humans cannot be expected to write correct syntax for columns containing embedded commas when using a comma delimited file. However, "|" delimited files work well because this character is not part of names or addresses. This tab delimited stuff is just hard to deal with.

#/usr/bin/perl
use strict;
use warnings;
$|=1;  #turns stdio buffer off so that error line order is correct

while (defined (my $line = <DATA>))
{
    chomp $line;                #removes end of line character(s)
    next unless $line =~ /\S/;  # skip blank lines
    
    my ($motherId,$yearBirth,$individualId) = (split /\t/,$line)[4,13,
+2];
    $yearBirth //= "";  # DATA line 6 is a bit weird
                        # this defines $yearBirth as "" if undefined
    
    print "UniqId=$individualId  Mother=$motherId  BirthY=$yearBirth \
+n";

}

=Prints
UniqId=1  Mother=3  BirthY=51 
UniqId=2  Mother=  BirthY= 
UniqId=3  Mother=36  BirthY=65 
UniqId=4  Mother=3  BirthY=50 
UniqId=5  Mother=3  BirthY=47 
UniqId=6  Mother=3  BirthY= 
UniqId=7  Mother=3  BirthY= 
UniqId=8  Mother=3  BirthY=42 
UniqId=9  Mother=3  BirthY=39 
UniqId=10  Mother=3  BirthY=35 
UniqId=11  Mother=8  BirthY=11 
UniqId=12  Mother=8  BirthY=9 
=cut


__DATA__
BC000    1    1    2    3    F          3    51                    51 
+   
BC000    0    2            M                                      999
BC000    0    3    37    36    F                                  65  
+  
BC000    0    4    2    3    M                                  50    
+50
BC000    0    5    2    3    F              45                    47  
+  46
BC000    0    6    2    3    F          3    42                       
+ 
BC000    0    7    2    3    M                                      99
+9
BC000    0    8    2    3    F          3                        42   
+ 
BC000    0    9    2    3    F          1                        39   
+ 
BC000    0    10    2    3    F          3                        35  
+  
BC000    0    11    45    8    M                                  11  
+  
BC000    0    12    45    8    F                                  9
[download]

#/usr/bin/perl
use strict;
use warnings;
$|=1;  #turns stdio buffer off so that error line order is correct

while (defined (my $line = <DATA>))
{
    chomp $line;                #removes end of line character(s)
    next unless $line =~ /\S/;  # skip blank lines
    
    my ($motherId,$yearBirth,$individualId,$what) = (split /\|/,$line)
+[4,13,2,14];
    $yearBirth //= "";  # DATA line 6 is a bit weird
                        # this defines $yearBirth as "" if undefined
    $what //= "";       # I don't know what this is?
    
    print "UniqId=$individualId  Mother=$motherId  BirthY=$yearBirth W
+hat=$what\n";

}

=Prints
UniqId=1  Mother=3  BirthY=51 What=
UniqId=2  Mother=  BirthY= What=999
UniqId=3  Mother=36  BirthY=65 What=
UniqId=4  Mother=3  BirthY=50 What=50
UniqId=5  Mother=3  BirthY=47 What=46
UniqId=6  Mother=3  BirthY= What=
UniqId=7  Mother=3  BirthY= What=999
UniqId=8  Mother=3  BirthY=42 What=
UniqId=9  Mother=3  BirthY=39 What=
UniqId=10  Mother=3  BirthY=35 What=
UniqId=11  Mother=8  BirthY=11 What=
UniqId=12  Mother=8  BirthY=9 What=
=cut


__DATA__
BC000|1|1|2|3|F  ||3|51|||||51|
BC000|0|2|||M  |||||||||999
BC000|0|3|37|36|F  ||||||||65|
BC000|0|4|2|3|M  ||||||||50|50
BC000|0|5|2|3|F  |||45|||||47|46
BC000|0|6|2|3|F  ||3|42||||||
BC000|0|7|2|3|M  |||||||||999
BC000|0|8|2|3|F  ||3||||||42|
BC000|0|9|2|3|F  ||1||||||39|
BC000|0|10|2|3|F  ||3||||||35|
BC000|0|11|45|8|M  ||||||||11|
BC000|0|12|45|8|F  ||||||||9|
[download]

#!/usr/bin/perl
use warnings;
use strict;
use Data::Dumper;
    
my $filename = 'pedigree_proband_testfile.txt';
my %n = (birth_year=> 13,  motherID=> 4,  individualID=> 2, gender => 
+5);
my %mominfo;

open my $f, "<", $filename or die "ERROR: Cannot open $filename:$!";

while (<$f>){
  chomp;
  my @c = split /\t/, $_; 
  my $currentMomID = $c[ $n{motherID} ]  || $c[ $n{individualID} ] or 
+die "No Mother ID in record nbr $.=$_;";
  my $currentMom = $mominfo{ $currentMomID } ||= {}; # Create an empty
+ mom if she does not exist
  if ($currentMomID == $c[ $n{individualID} ]){
    print "This is Mom ($currentMomID)'s main record: $_\n";
    $currentMom->{birth_year} = $c[ $n{birth_year} ];
  }else{
    push @{ $currentMom->{CHILDREN} }, {ID=>  $c[ $n{individualID} ] ,
+ 
                                        birth_year =>  $c[ $n{birth_ye
+ar} ],
                                        gender => $c[ $n{gender} ]};
  }
}

print Dumper \%mominfo;
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