Q:

This divides the class of game objects into material and immaterial

The fact is that a material object must have a field that does not make sense to an immaterial object
Here is a class whose objects I want to divide into material and immaterial:
public class GameObject {
    protected boolean isHidden;
    protected Coordinate position;
    protected int pictureWidth, pictureHeight;
    public Object filling;

    protected GameObject(int pictureWidth, int pictureHeight, Coordinate position, boolean isHidden, Color filling){
        this.position = position;
        this.pictureWidth = pictureWidth;
        this.pictureHeight = pictureHeight;
        this.isHidden = isHidden;
        this.filling = filling;
    }

    protected GameObject(Coordinate position, boolean isHidden, BufferedImage filling){
        this.position = position;
        this.pictureWidth = filling.getWidth();
        this.pictureHeight = filling.getHeight();
        this.isHidden = isHidden;
        this.filling = filling;
    }

    protected GameObject(GameObject gameObject) throws IOException{
        this.position = new Coordinate(gameObject.getPosition());
        this.pictureWidth = gameObject.getPictureWidth();
        this.pictureHeight = gameObject.getPictureHeight();
        this.isHidden = gameObject.isHidden();
        this.filling = gameObject.getFilling(); // передается ссылка
    }

    private GameObject getGameObject() { 
        return this;
    }
    public void paint(Graphics gr) throws IOException{
        if(filling instanceof BufferedImage) {
            gr.drawImage((Image) filling, position.getX(), position.getY(), null);
        }
        else if(filling instanceof Color) {
            gr.setColor((Color) filling);
            gr.fillRect(position.getX(), position.getY(), pictureWidth, pictureHeight);
        } else {
            System.err.println("programmer, you forgot to add a way to draw this object");
        }
    }
    Object getFilling() {
        return filling;
    }

    int getPictureWidth() {
        return pictureWidth;
    }
    int getPictureHeight() {
        return pictureHeight;
    }
    boolean isHidden() {
        return isHidden;
    }
    Coordinate getPosition() {
        return position;
    }
}

I suppose that you need to create 2 classes that inherit GameObject, and make the constructor of the GameObject class visible only for these two classes:
1) class MaterialGameObject:
 public class MaterialGameObject extends GameObject{
    final int materialHeight;

    public MaterialGameObject(int materialHeight, int pictureWidth, int pictureHeight, Coordinate position, boolean isHidden, Color filling){
        super(pictureWidth, pictureHeight, position, isHidden, filling);
        this.materialHeight = materialHeight;
    }

    public MaterialGameObject(int materialHeight, Coordinate position, boolean isHidden, BufferedImage filling){
        super(position, isHidden, filling);
        this.materialHeight = materialHeight;
    }

    public MaterialGameObject(MaterialGameObject materialGameObject) throws IOException{
        super(materialGameObject.getGameObject());
        this.materialHeight = materialGameObject.getMaterialHeight();
    }

    public int getMaterialHeight() {
        return materialHeight;
    }

}

class ImMaterialGameObject:
public class ImMaterialGameObject extends GameObject{

    public ImMaterialGameObject(int pictureWidth, int pictureHeight, int materialHeight, Coordinate position, boolean isHidden, Color filling){
        super(pictureWidth, pictureHeight, position, isHidden, filling);
    }

    public ImMaterialGameObject(Coordinate position, boolean isHidden, BufferedImage filling){
        super(position, isHidden, filling);
    }

    public ImMaterialGameObject(GameObject gameObject) throws IOException{
        super(gameObject);
    }

}

Do you know a better way to solve the problem?

A:

getGameObject - Never used. You should remove it. You can always use this, no need to have a function. If you meant for this to be protected, still remove it. Again use the object, no need to have a function.
ImMaterialGameObject - Unless "Im" stands for something, Immaterial is one word.
public Object filling; - You've obviously noticed all classes in Java extend from Object. Taking advantage of this is hackey. Instead you could use 2 separate fields and check if one or the other is null.
Onto the question "Is there a better approach?"
Without knowing what you're using this for, I'd say there's nothing wrong with your approach. To make things more clear, set GameObject to abstract. 

