The following code updates the count 'in place' using a regex. If the regex fails an initial count is generated.

use strict;
use warnings;

open FLASHCOUNT, '<', '/var/www/html/log/FLASHCOUNT.txt';
my $line = <FLASHCOUNT> || '';
close(FLASHCOUNT);

$line = 'count = 1' unless $line =~ s/(\d+)/$1 + 1/e;

open FLASHCOUNT, '>', '/var/www/html/log/FLASHCOUNT.txt';
print FLASHCOUNT $line;
close FLASHCOUNT;

print "Content-type: text/html\n\n";
print $line;
print "</BODY>\n";
print "</HTML>";
[download]

Update: note ikegami's comment re race conditions.