Turning query errors to Exceptions in MySQLi

Some time ago I managed to sort this matter out. As it was pointed out in the other answer,

mysqli_report(MYSQLI_REPORT_ERROR | MYSQLI_REPORT_STRICT);

is a correct way to tell mysqli to throw exceptions.

Just make sure you don't wrap every query in a try-catch. This is a very common misconception that as soon as you started using exceptions you should start throwing tries and catches left and right. Quite contrary, try-catch should be used warily. While 99% of your errors shouldn't be handled in place, but rather by a site-wide error handler. You may read more on the topic from my article on PHP error reporting

do I have to check if mysqli_query() returned false?

No.

You should be able to do what you require and instruct the mysqli driver to throw exceptions on SQL errors, but you will need to enable MYSQLI_REPORT_ERROR if it is not already....

mysqli_report(MYSQLI_REPORT_ERROR | MYSQLI_REPORT_STRICT)

mysqli_query() should now throw exceptions on error. You do not need to check the return value for failure (which won't happen anyway because an exception is thrown).

public function mysqlQuery($SQL) {
    try {
        $this->Result = mysqli_query($this->DBlink, $SQL);
    } catch (mysqli_sql_exception $e) {
        throw new MySQLiQueryException($SQL, $e->getMessage(), $e->getCode());
    }
    return $this->Result;
}

(NB: I changed $this->SQL to $SQL in the re-thrown exception.)