JavaScript Good Patterns - Is this a good example?
Lets look at a couple of improvements:

new is creating an object that you aren't using. read edit for clarification

new window.APP_NAME.moduleName.functionInsideModule1({
    name: 'Mary'
});


The function functionInsideModule1 doesn't return an object and doesn't assign anything to this. So there is no need for the new. 
You're referencing the whole namespace window.APP_NAME.moduleName every time. Instead try putting it in a variable. Or maybe on the inside of the module  refer to this instead.

(function () {

    var moduleName = window.APP_NAME.moduleName;

    moduleName.functionInsideModule1 = function (options) {
        /* snip ... */

        moduleName.globalVariableAvailableForEveryModuleFunctions += 1;
        // OR
        this.globalVariableAvailableForEveryModuleFunctions += 1;
    };

    moduleName.functionInsideModule1({
         name: 'Mary'
    });

}());


Edit:-

To create a new instance of moduleName you need to do this:

(function () {
    // private variables go here
    var globalPrivateCount = 0;

    var moduleName = window.APP_NAME.moduleName = window.APP_NAME.moduleName || function moduleName(initialval) {
        globalPrivateCount ++;

        this.globallyAccessible = 'unique for each instance';

        this.functionInsideModule1 = function (options) { ... };
        this.functionInsideModule2 = function (options) { ... };

        var instancePrivateVariable = initialVal;

        this.incr = function () { instancePrivateVariable++; };

        return this;
    };
    moduleName.nonUniqueGloballyAccessible = 0;
}());


and the usage is something like:

var moduleName = window.APP_NAME.moduleName;
var firstInstance = new moduleName(10);
// OR
var anotherInstance = new window.APP_NAME.moduleName(20);