Try counting slashes and prune:

#!/usr/bin/perl
# linux only
use warnings;
use strict;
use File::Find;
use File::Spec;

if (@ARGV < 2){print "Usage: $0 dir depth\n";exit}
my ($path, $depth)= @ARGV;

my $abs_path = File::Spec->rel2abs($path); #in case you enter . for di
+r
my $m = ($abs_path) =~ tr!/!!;  #count slashes in top path

find (\&found,$abs_path);
exit;

sub found{
 my $n = ($File::Find::name) =~ tr!/!!; #count slashes in file
   return $File::Find::prune = 1 if $n > ($m + $depth);
   # do stuff here.
   #print "$_\n";                #name only
   print "$File::Find::name\n";  #name with full path
}
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or use File::Find::Rule

#!/usr/bin/perl
use warnings;
use strict;
use File::Find::Rule;

my $word = shift || 'perl';

# find all the files of a given directory
my $directory = shift || '/home/zentara';
my $depth = shift || 3;
my $rule1 = File::Find::Rule->new;
$rule1->maxdepth($depth);
$rule1->file;
$rule1->grep(qr/\Q$word\E/);
#$rule->name( '*.pm' );
my @files = $rule1->in($directory);
print "@files\n";
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