Java: Ternary with no return. (For method calling)

No, you can't. But what's the point of this over an if-else statement? Are you really trying to save 7 characters?

if (name.isChecked()) {
    name.setChecked(true);
} else {
    name.setChecked(false);
}

or if you prefer bad style:

if (name.isChecked()) name.setChecked(true); else name.setChecked(false);

Never mind the fact that you can just do (in this case):

name.setChecked(name.isChecked());

The point of the ternary or "conditional" operator is to introduce conditionals into an expression. In other words, this:

int max = a > b ? a : b;

is meant to be shorthand for this:

int max;
if ( a > b ) {
    max = a;
} else {
    max = b;
}

If there is no value being produced, the conditional operator is not a shortcut.

I was wondering if it was possible to do a ternary operation but without returning anything.

No it is not possible:

1. The 2nd and 3rd operands are required to be non-void expressions; i.e. they must produce some actual value.

   "It is a compile-time error for either the second or the third operand expression to be an invocation of a void method." - JLS 15.25.

2. A ternary expression is an expression, and cannot be used as a statement.

   "Certain kinds of expressions may be used as statements by following them with semicolons." ... and the ternary expression is not one of those kinds - JLS 14.8.

If you really, really want to use a ternary expression but not use the value of the expression, then the simplest thing is to assign the value to a dummy variable, and add an annotation to suppress the warning about the variable not being used.

But a better idea is to use a plain if statement.

If it's not possible in Java is it possible in other languages, if so which ones apply?

I'm a bit rusty, but I believe that C, C++ and Perl all allow arbitrary expressions to be used in places where their values are not used.