I'm creating a Qt application to operate on Home windows, Mac and Linux (Ubuntu Desktop). Development is within C++ obviously and it is a desktop application that must store and retrieve "tabluar" info on a continuing basis.

I don't want to bother my customers with pre-setting up MySQL or anything alike.

It is possible to lightweight database engine I'm able to use being an SDK from the application that will store in your area around the file system and support SQL queries?

Try SQLite : http://sqlite.org

In my opinion that this (SQLite) is going to do everthing that the asking inside your question. From the own go through it work half way decent for small datasets. but when you're utilizing it for big levels of data, you might want to think about a full blow sql server