Fun puzzle. Here is my kind of hybrid answer. 4 nested loops + some regex to weed out straight lines.

use strict;
use warnings;

my $data = {
   A => [qw/B C D E F I G J/],
   B => [qw/A C D E F H J I/],
   C => [qw/A B F I D E/],
   D => [qw/A C B G J E/],
   E => [qw/A D C B G H I J/],
   F => [qw/A C B I H J/],
   G => [qw/D A E H I J/],
   H => [qw/F B I J G E/],
   I => [qw/B F C A H G E J/],
   J => [qw/A B F H I E/],
};
my $line = qr/[ACFI]{3}|[ADGJ]{3}|[BCDE]{3}|[BFHJ]{3}|[EGHI]{3}/;

my $triangles = {};

for my $p1 (keys %$data)  {
    for my $p2 (@{$data->{$p1}}) {
        for my $p3 (@{$data->{$p2}}) {
            next if $p3 eq $p1;
            for my $p4 (@{$data->{$p3}}) {
              if ($p4 eq $p1) {
              my $tri = join( "", sort @{[$p1,$p2, $p3]});
              $triangles->{$tri}++ unless $tri =~ $line;
              }
            }
        }
    }
}
print $_,"\n" for sort keys %$triangles;
print 0+keys %$triangles;
[download]