Demo: http://jsfiddle.net/ThinkingStiff/y9K9Y/
If the <div> you’re searching for has a class, you can use getElementsByClassName():
document.getElementById( 'parentDiv' ).getElementsByClassName( 'childDiv' )[0];
If it doesn’t have a class you can use getElementsByTagName():
document.getElementById( 'parentDiv' ).getElementsByTagName( 'div' )[0];
And if it has an id you can, of course, just use getElementById() to find it no matter where it is in the DOM:
document.getElementById( 'childDiv' );