JavaFX: Взаимодействие двух окон

Вам необходимо определить соответствующие поле во втором контроллере, для передачи ему объекта.

import javafx.application.Application;
import javafx.event.ActionEvent;
import javafx.scene.Scene;
import javafx.scene.control.Button;
import javafx.scene.control.ButtonType;
import javafx.scene.control.Dialog;
import javafx.scene.layout.StackPane;
import javafx.scene.layout.VBox;
import javafx.stage.Stage;
import javafx.util.Callback;
import java.util.Optional;
public class App extends Application {
    @Override
    public void start(Stage primaryStage) throws Exception {
        RootController root = new RootController(); // or fxml load
        primaryStage.setScene(new Scene(root, 100, 100));
        primaryStage.show();
    }
    private static class RootController extends StackPane {
        private CustomObject customObject = new CustomObject();
        // or fxml initialization
        public RootController() {
            Button bShowDialog = new Button("Show");
            bShowDialog.setOnAction(this::show);
            getChildren().add(bShowDialog);
        }
        private void show(ActionEvent event) {
            DialogController content = new DialogController(); // or fxml load
            content.customObject = customObject;
            Dialog<Boolean> dialog = new Dialog<>();
            dialog.getDialogPane().setContent(content);
            dialog.setResultConverter(content.resultConverter);
            dialog.getDialogPane().getButtonTypes().addAll(ButtonType.OK, ButtonType.CANCEL);
            Optional<Boolean> result = dialog.showAndWait();
            if ( result.isPresent() ) {
                System.out.println("Result = " + result.get());
                System.out.println("New value = " + customObject.i);
            }
        }
    }
    private static class DialogController extends VBox {
        CustomObject customObject;
        // or fxml initialization
        public DialogController() {
            Button bInc = new Button("Inc");
            bInc.setOnAction(this::inc);
            Button bDec = new Button("Dec");
            bDec.setOnAction(this::dec);
            getChildren().addAll(bInc, bDec);
        }
        private void inc(ActionEvent event) { customObject.i++; }
        private void dec(ActionEvent event) { customObject.i--; }
        Callback<ButtonType, Boolean> resultConverter = param -> param == ButtonType.OK && customObject.i > 0;
    }
    private static class CustomObject {
        int i;
    }
}