What exactly isn't working? Are you not getting any results, or are the prepare/execute methods failing?

When you use the '?' placeholder, you don't need to quote your arguments--the placeholder handles the quoting for you. Perhaps this is the problem?

use DBI;

my $dbh = DBI->connect('foo', 'bar', 'baz', 'Oracle')
    or die "Can't connect: ", $DBI::errstr;

my $sth = $dbh->prepare("select first_name from people where last_name
+ = ?")
    or die "Can't prepare: ", $dbh->errstr;

my $last_name = "Bar";
$sth->execute($last_name);

my $first_name;
$sth->bind_columns(undef, \$first_name);
while ($sth->fetch) {
    print $first_name, "\n";
}
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You need to use place holders of the form ":<whatever>"

        ### Prepare weeks in month cursor
        $weeks_in_mth_sth = 
            $dbh->prepare( qq {
           SELECT  count(*)
             FROM  year_month_weeks
             WHERE year_no = :year
               AND month_no = :month
                          }

Then you need to 'bind' the place holders to an actual variable as below :

    $weeks_in_mth_sth->bind_param(':year', $p_year);
    $weeks_in_mth_sth->bind_param(':month', $p_month);

Then you should be able to execute your query :

    $weeks_in_mth_sth->execute
        or die "Can't execute SQL statement: $DBI::errstr Stopped\n";

Good luck !