Data not insert and update through ajax and jQuery in admin page?

I see that there’s an internal server error (see screenshot) when you tried to update a record in the database, and the error is likely because you incorrectly called $wpdb->update() which has the syntax of — and note that the first three parameters ($table, $data and $where) are required:

wpdb::update( string $table, array $data, array $where, array|string $format = null, array|string $where_format = null )

But in your code, you did not set the third parameter ($where):

// In update_records():
$db_updated = $wpdb->update( $wpdb->prefix.'faqq',
    array( 'question' => $question,
           'answer'   => $answer, array( 'ID' => $id ) )
);

And I guess the array( 'ID' => $id ) is the $where part, but you probably made a typo which resulted in that part being part of the second parameter instead.

So fix that and your code would work properly. Example:

$db_updated = $wpdb->update( $wpdb->prefix.'faqq',
    array( 'question' => $question,
           'answer'   => $answer,
    ),
    array( 'ID' => $id )
);

As for your insert_records() function, it looks fine to me, so there shouldn’t be an issue when inserting new records to your database table.

Additionally, you should also (later) do things like data sanitization and call wp_die() to end the AJAX request, and fix the issue as seen here.