is it possible to query a database and display the result in an alert box with php

You can try something like this:

$.get( "test.php" );

if you need send data into your php function:

$.get( "test.php", { name: "John", time: "2pm" } );

if you need to write data that you get into alert window

$.get( "test.php", function( data ) {
  alert( "Data Loaded: " + data );
});

those are examples from here:
https://api.jquery.com/jquery.get/[^]

If you don't want to read it all just scroll down to examples a try it with examples.

However, if you want to understand how it works read the documentation.

And also this could be useful if you want do more complex things than get.
https://api.jquery.com/jQuery.ajax/[^]