First of all, note that by writing $delta->{'months'} you are reaching into the internals of the object, which bypasses any accessors and may break if the field names happen to change. So don't do that and use only the accessors instead. Second, note the section "Math Methods" in DateTime and the documentation of DateTime::Duration: Because the number of days per month is not fixed, IMHO it's better to tell the modules exactly what kind of math you want to do. For instance, in your case, you might want the ->delta_md method. Note that it returns an absolute difference, which is why an additional check of $date>$now is necessary, but personally I don't mind being explicit.

use warnings;
use strict;
use DateTime;
use DateTime::Format::Human::Duration;

my $now = DateTime->new( time_zone => "UTC",
    year => 2019, month => 6, day => 10 );

my $fmt = DateTime::Format::Human::Duration->new();

for my $mon ( 4, 6, 8 ) {
    for my $day ( 1, 9 , 10, 11, 19 ) {
        my $date = DateTime->new( time_zone => "UTC",
            year => 2019, month => $mon, day => $day );
        my $delta = $now->delta_md($date);
        my ($months,$days) = $delta->in_units(qw/ months days /);
        
        print $now->ymd, " to ", $date->ymd, " is ",
            $fmt->format_duration($delta,
                units=>[qw/months days/]); 
        
        if ( $date>$now && ( $months>2 || $months==2 && $days>0 ) ) {
            print " *** Alert ***";
        }
        print "\n";
    }
}

__END__

2019-06-10 to 2019-04-01 is 2 months and 9 days
2019-06-10 to 2019-04-09 is 2 months and 1 day
2019-06-10 to 2019-04-10 is 2 months
2019-06-10 to 2019-04-11 is 1 month and 29 days
2019-06-10 to 2019-04-19 is 1 month and 21 days
2019-06-10 to 2019-06-01 is 9 days
2019-06-10 to 2019-06-09 is 1 day
2019-06-10 to 2019-06-10 is no time
2019-06-10 to 2019-06-11 is 1 day
2019-06-10 to 2019-06-19 is 9 days
2019-06-10 to 2019-08-01 is 1 month and 21 days
2019-06-10 to 2019-08-09 is 1 month and 29 days
2019-06-10 to 2019-08-10 is 2 months
2019-06-10 to 2019-08-11 is 2 months and 1 day *** Alert ***
2019-06-10 to 2019-08-19 is 2 months and 9 days *** Alert ***
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(Of course it's also possible to do this in SQL.)