Employing a little regex look ahead trickery gives something like this:

use strict;
use warnings;

my $pat = 'ono';
my $str = <<STR;
ono is the pattern to match. ono is found once in onono
STR
#123456789012345678901234567890123456789012345678901234
#         1         2         3         4         5    

for my $count (1..4) {
    if ($str =~ /(?:(?:(?!$pat).)*($pat)){$count}/) {
        print "Match at $-[1] for count = $count\n";
    } else {
        print "No match for count = $count\n";
    }
}
[download]

Prints:

Match at 0 for count = 1
Match at 29 for count = 2
Match at 50 for count = 3
No match for count = 4
[download]

Note that at least simple regex matches are fine in the pattern including look ahead and look back matches, but avoid capture groups.