Is it possible to pass SQL queries directly to a database that is hidden behind a firewall and any other security mechanism?

Since your website needs to be public, security mechanisms will allow public web traffic to communicate with your web application/s (generally over port 80/443). The web application has open access to the database in order to return (update) the requested (changed) information.
The Firewalls and similar intrusion detection mechanisms provide little or no defense against full-scale SQL Injection web attacks.
In SQL Injection, the hacker uses SQL queries and creativity to get to the database of sensitive corporate data through the web application.
SQL or Structured Query Language is the computer language that allows you to store, manipulate, and retrieve data stored in a relational database (or a collection of tables which organise and structure data). SQL is, in fact, the only way that a web application (and users) can interact with the database. Examples of relational databases include Oracle, Microsoft Access, MS SQL Server, MySQL, and Filemaker Pro, all of which use SQL as their basic building blocks.
SQL commands include SELECT, INSERT, DELETE and DROP TABLE. DROP TABLE is as ominous as it sounds and in fact will eliminate the table with a particular name.
In the legitimate scenario of the login page example above, the SQL commands planned for the web application may look like the following:
SELECT count(*)
FROM users_list_table
WHERE username=’FIELD_USERNAME’
AND password=’FIELD_PASSWORD”
In plain English, this SQL command (from the web application) instructs the database to match the username and password input by the legitimate user to the combination it has already stored.
Each type of web application is hard coded with specific SQL queries that it will execute when performing its legitimate functions and communicating with the database. If any input field of the web application is not properly sanitised, a hacker may inject additional SQL commands that broaden the range of SQL commands the web application will execute, thus going beyond the original intended design and function.
A hacker will thus have a clear channel of communication (or, in layman terms, a tunnel) to the database irrespective of all the intrusion detection systems and network security equipment installed before the physical database server.