Instead of comparing each dest of each module with each dest of any other module, just create a hash of dests and store which modules use it. Then, if the resulting hash has more than one key for a dest, they correspond to the connected modules. I stored the indices in the inner hashes for easier retrieval of the corresponding sources:

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my %h = ('module2(' => {source => ['.akjsdk', '.qahwgd'] ,
                        dest   => ['ajhsx', 'uiahs'] },
         'module3(' => {source => ['.x', '.y'] ,
                        dest   => ['z', 'uiahs'] },
         'module1(' => {source => ['.akjsdk1', '.sjkdhs11', '.qahwgd1'
+],
                        dest   => ['ajhsx', 'ajhdj', 'uiahs']});

my %connection;
for my $module (keys %h) {
    for my $dest_index (0 .. $#{ $h{$module}{dest} }) {
        my $dest = $h{$module}{dest}[$dest_index];
        undef $connection{$dest}{$module}{$dest_index};
    }
}
for my $dest (keys %connection) {
    if ((my @modules = keys %{ $connection{$dest} }) > 1) {
        say "Connected:";
        for my $module (@modules) {
            say "\t$module $h{$module}{source}[$_]" for keys %{ $conne
+ction{$dest}{$module} };
        }
    }
}
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