You are preparing two different queries within loops. This can be tricky because some databases don't release the cursor of the query, thus running out of cursors after a while (been there and seen it).
While the loop is inevitable for the first query, you can and should put the second (update) query outside the loop:

my $insert_query = $cdbh->prepare("insert into keyword (zkey, value) v
+alues (?, ?)";

while ($csth->fetch)
{
    $query = "select $primarykey, $fieldname from $tablename";

    # this sth uses the ODBC $dbh for the first three columns, but
    #    tries to use the SQLite $cdbh for the 4th column
    my $sth = $::dbh->prepare($query);
    $sth->execute;
    my ($key, $value);
    $sth->bind_columns(\$key, \$value);
    
    while ($sth->fetch)
    {
        next if !$key;
        next if !$value;
            
        $value = lc($value);
        my @words = split(/\W/, $value);
            
        for (my $i = 0; $i < scalar(@words); $i++)
        {
            my $word = $words[$i - 1];
            next if !$word;
                
            # trim
            $word =~ s|^ *(.+) *$|$1|;
            next if !$word;
                
            $insert_query->execute("$tablename.$key", $word)";);
        }
    }
}
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Note that using $dbh->quote is not neccessary when you use placeholders. Also note that this is untested.