There were a lot of problems with your code. I've tried to clean it up (but didn't test it beyond perl -c). Let me know if it doesn't do what you want.

The only halfway tricky thing my code does is that it avoids computing differences between query and base more than once by keeping a "sliding window" of differences in @r; the new element is always added to $r[ $i % 2 ], and then all elements are tested to make sure that they are not greater than $j. This can be optimized a lot more, at the expense of changing the structure of the code more than I wanted.

#!/usr/bin/perl

# This is a script which compares the query file and base with some
# threshold given by the user

use strict;
use warnings;

$| = 1;
print "Enter the Threshold Value:  \n";
my $j = <>;
chomp $j;

my $query_data = read_data( '10.asc.output' );

my $dir = '/home/users/dachs/perl_scripts/perlup'; #dir path where
opendir my $dh, $dir or die "System can't open directory for reading\n
+";

while (my $filename = readdir $dh) {
  next unless $filename =~ /\.asc.output$/i;
  my $base_data = read_data( $filename );

  open my $out, ">$filename.cmp"
    or die "cannot open file for writing: $!"; # Creates an output fil
+e

  my @r = map abs( $query_data->[ $_ ] - $base_data->[ $_ ] ), 0, 1;
  my $i = 2;
  while ( $i < @$query_data ) {
    $r[ $i % 3 ] = abs( $query_data->[ $i ] - $base_data->[ $i ] );
    if ( $r[ 0 ] <= $j && $r[ 1 ] <= $j && $r[ 2 ] <= $j ) {
      my $matched = join "\t", @{ $query_data }[ $i-2..$i ];
      print $out "Matched: $matched\n";
    }
    $i++;
  }
  close $out;
}
closedir $dh;

sub read_data {
  my $filename = shift;
  open my $in, $filename or die;
  my @data;
  while ( <$in> ) {
    push @data, $1 if /(\S+)\s+\S+/; # difficult to advise on regex
                                     # without knowing the details of
                                     # the data
  }
  close $in;
  return \@data;
}
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