waswas-fng's idea is the first thing I would try; something like this:

my $max_line_count = $whatever;
my $file_count = my $line_count = 0;
my $outfh;

sub printAndCount
{
    local $_ = shift;
    my $lines2print = tr/\n/\n/;  # simple way to count \n's
    if ( $file_count == 0 or 
         $line_count + $lines2print > $max_line_count )
    {
        close $outfh if ( $file_count );
        open $outfh, sprintf( ">track%03d.txt", ++$file_count ) or die
+ "open failed: $!";
    }
    print $outfh;
    $line_count += $lines2print;
}
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Note that this idea doesn't exactly match what you said:

The end goal is that when a cetain number of lines is reached i want to close the file handle, and open a new one with the same name that points to a different file.

You can't "open a different file" using the same file name over and over. If you don't use a different file name each time, you'll simply be deleting the last set of lines that you wrote as soon as you open a "new" file.