Finding the height is in fact no different for N-ary trees than for any other type of tree. A leaf node has height 0, and a non-leaf node has height one more than the tallest of its children.
Have a recursive function that in pseudocode does this in Java:
public static int getHeight(Node n){
if(n.isLeaf()){
return 0;
}else{
int maxDepth = 0;
foreach(Node child : n.getChildren()){
maxDepth = Math.max(maxDepth, getHeight(child));
}
return maxDepth + 1;
}
}