JavaFX – How to use a method in a controller from another controller?

The “quick and dirty” way is to give the Stage1Controller a reference to the Stage2Controller:

public class Stage1Controller {

    private final Stage2Controller stage2Controller ;

    public void setStage2Controller(Stage2Controller stage2Controller) {
        this.stage2Controller = stage2Controller ;
    }

    // ...

    @FXML 
    private void handleTranslate (ActionEvent event){
        translateFirstStage();
        stage2Controller.setLabel(...);
    }

    // other code as before ...
}

Now in your main app:

public class MainApp  extends Application {

    private Stage1Controller stage1Controller ;
    private Stage2Controller stage2Controller ;

    @Override
    public void start(Stage primaryStage) {
        firstStage();
        secondStage();

        stage1Controller.setStage2Controller(stage2Controller);

        // ...
    }

    public void firstStage() {
        try {
            // Load root layout from fxml file.
            FXMLLoader loader = new FXMLLoader();
            loader.setLocation(Main.class.getResource("view/stage1.fxml"));
            rootLayout = (BorderPane) loader.load();

            stage1Controller = loader.getController();

            // Show the scene containing the root layout.
            Scene scene = new Scene(rootLayout);
            primaryStage.setScene(scene);
            primaryStage.show();
        } catch (IOException e) {
            e.printStackTrace();
        }
    }

    public void secondStage() {

        try {
            // Load root layout from fxml file.
            FXMLLoader loader = new FXMLLoader();
            loader.setLocation(Main.class.getResource("view/stage2.fxml"));
            XD = (AnchorPane) loader.load();

            stage2Controller = loader.getController();

            // Show the scene containing the root layout.
            Scene scene = new Scene(XD);
            Stage stage = new Stage();
            stage.setScene(scene);
            stage.show();
        } catch (IOException e) {
            e.printStackTrace();
        }
    }

    // ...
}

A more robust approach to this, though, is to let both controllers access a shared model class, storing the data. If you represent the data using JavaFX observable properties, the controllers can listen for changes on the properties they care about. For example:

public class Model {

    private final StringProperty text = new SimpleStringProperty("Initial text...");

    public StringProperty textProperty() {
        return text ;
    }

    public final void setText(String text) {
        textProperty().set(text);
    }

    public final String getText() {
        return textProperty().get();
    }

    // other properties as needed...
}

Now your controllers will look like this:

public class Stage1Controller {

    private Model model ;

    public void setModel(Model model) {
        this.model = model ;
    }

    @FXML 
    private void handleTranslate (ActionEvent event){
        translateFirstStage();

        model.setText(...); // value will appear in stage2 controller's label!
    }

    // ...
}

and

public class Stage2Controller {

    @FXML
    private Label lb ;

    private Model model ;

    public void setModel(Model model) {
        lb.textProperty().unbind();
        this.model = model ;
        lb.textProperty().bind(model.textProperty());
    }

    // ...
}

And in this case your main app looks like:

public class MainApp extends Application {

    private final Model = new Model();

    @Override
    public void start(Stage primaryStage) {
        // ...
    }

    public void firstStage() {
        try {
            // Load root layout from fxml file.
            FXMLLoader loader = new FXMLLoader();
            loader.setLocation(Main.class.getResource("view/stage1.fxml"));
            rootLayout = (BorderPane) loader.load();

            Stage1Controller controller = loader.getController();
            controller.setModel(model);

            // Show the scene containing the root layout.
            Scene scene = new Scene(rootLayout);
            primaryStage.setScene(scene);
            primaryStage.show();
        } catch (IOException e) {
            e.printStackTrace();
        }
    }

    public void secondStage() {

        try {
            // Load root layout from fxml file.
            FXMLLoader loader = new FXMLLoader();
            loader.setLocation(Main.class.getResource("view/stage2.fxml"));
            XD = (AnchorPane) loader.load();

            Stage2Controller controller = loader.getController();
            controller.setModel(model);

            // Show the scene containing the root layout.
            Scene scene = new Scene(XD);
            Stage stage = new Stage();
            stage.setScene(scene);
            stage.show();
        } catch (IOException e) {
            e.printStackTrace();
        }
    }
}