There are two problems:

1. Java uses pass by value, not by reference. Changing the reference inside a method won't be reflected into the passed-in reference in the calling method.
2. Integer is immutable. There's no such method like Integer#set(i). You could otherwise just make use of it.

To get it to work, you need to reassign the return value of the inc() method.

integer = inc(integer);

To learn a bit more about passing by value, here's another example:

public static void main(String... args) {
    String[] strings = new String[] { "foo", "bar" };
    changeReference(strings);
    System.out.println(Arrays.toString(strings)); // still [foo, bar]
    changeValue(strings);
    System.out.println(Arrays.toString(strings)); // [foo, foo]
}
public static void changeReference(String[] strings) {
    strings = new String[] { "foo", "foo" };
}
public static void changeValue(String[] strings) {
    strings[1] = "foo";
}

You can try using org.apache.commons.lang.mutable.MutableInt from Apache Commons library. There is no direct way of doing this in the language itself.

Most of the classes such as Integer that derive from Java's abstract Number class are immutable., i.e. once constructed, they can only ever contain that particular number.

A useful benefit of this is that it permits caching. If you call:

Integer i = Integer.valueOf(n);

for -128 <= n < 127 instead of:

Integer i = Integer.new(n)

you get back a cached object, rather than a new object. This saves memory and increases performance.

In the latter test case with a bare int argument, all you're seeing is how Java's variables are passed by value rather than by reference.

You have several choices. The one that makes the most sense really depends on what you're trying to do.

Choice 1: make toyNumber a public member variable in a class

class MyToy {
  public int toyNumber;
}

then pass a reference to a MyToy to your method.

void play(MyToy toy){  
    System.out.println("Toy number in play " + toy.toyNumber);   
    toy.toyNumber++;  
    System.out.println("Toy number in play after increement " + toy.toyNumber);   
}

Choice 2: return the value instead of pass by reference

int play(int toyNumber){  
    System.out.println("Toy number in play " + toyNumber);   
    toyNumber++;  
    System.out.println("Toy number in play after increement " + toyNumber);   
    return toyNumber
}

This choice would require a small change to the callsite in main so that it reads, toyNumber = temp.play(toyNumber);.

Choice 3: make it a class or static variable

If the two functions are methods on the same class or class instance, you could convert toyNumber into a class member variable.

Choice 4: Create a single element array of type int and pass that

This is considered a hack, but is sometimes employed to return values from inline class invocations.

void play(int [] toyNumber){  
    System.out.println("Toy number in play " + toyNumber[0]);   
    toyNumber[0]++;  
    System.out.println("Toy number in play after increement " + toyNumber[0]);   
}