Simple example showing a popup shown from a background thread (simulating receiving messages from a server...). (No fancy UI, just showing the idea).

PopupFromBackgroundThreadDemo.java

import java.util.Random;
import java.util.concurrent.Executor;
import java.util.concurrent.Executors;
import java.util.logging.Level;
import java.util.logging.Logger;

import javafx.application.Application;
import javafx.application.Platform;
import javafx.event.ActionEvent;
import javafx.event.EventHandler;
import javafx.geometry.Pos;
import javafx.scene.Scene;
import javafx.scene.control.Button;
import javafx.scene.control.Label;
import javafx.scene.control.TextField;
import javafx.scene.layout.VBox;
import javafx.stage.Popup;
import javafx.stage.Stage;
import javafx.stage.Window;

public class PopupFromBackgroundThreadDemo extends Application {
    
    private final Executor exec = Executors.newCachedThreadPool(runnable -> 
        new Thread(runnable){{ setDaemon(true);}});
    
    private final Logger logger = Logger.getLogger(getClass().getName());

	@Override
	public void start(Stage primaryStage) {
	    Popup popup = new Popup();
	    
	    Runnable messageReadSimulator = () -> {
	        
	        Random rng = new Random();
	        
	        while (true) {
	            try {
	                Thread.sleep(rng.nextInt(5000)+5000);
	            } catch (InterruptedException exc) {
	                logger.log(Level.WARNING, "Interruption to background thread", exc);
	                break ;
	            }
	        
	            showMessage("The server sent a message", popup, primaryStage);
	        
	        }
	    };
	    
	    exec.execute(messageReadSimulator);
	    
	    TextField textField = new TextField();
            Button button = new Button("Show message");
	    
	    EventHandler<ActionEvent> handler = e -> {
	        showMessage(textField.getText(), popup, primaryStage);
	        textField.setText("");
	    };
        
	    textField.setOnAction(handler);        
            button.setOnAction(handler);
        
            VBox root = new VBox(10, textField, button);
            root.setAlignment(Pos.CENTER);
	    
	    Scene scene = new Scene(root, 250, 100);
	    primaryStage.setScene(scene);
	    primaryStage.show();
	}
	
	public void showMessage(String message, Popup popup, Window owner) {
	    
	    logger.info("showMessage(...) called from thread "+Thread.currentThread()+". "
	            + "FX Application Thread? "+Platform.isFxApplicationThread());
	    
	    if (Platform.isFxApplicationThread()) {
	        _showMessage(message, popup, owner);
	    } else {
	        Platform.runLater(() -> _showMessage(message, popup, owner));
	    }
	}
	
	private void _showMessage(String message, Popup popup, Window owner) {

            logger.info("_showMessage(...) called from thread "+Thread.currentThread()+". "
                + "FX Application Thread? "+Platform.isFxApplicationThread());

	    if (! Platform.isFxApplicationThread()) {
	        throw new Error("Not on FX Application Thread");
	    }
	    popup.getContent().clear();
	    Label label = new Label(message);
	    Button button = new Button("OK");
	    button.setOnAction(e -> popup.hide());
	    VBox vbox = new VBox(10, label, button);
	    vbox.setStyle("-fx-background-color: antiquewhite; -fx-padding: 10;");
	    vbox.setAlignment(Pos.CENTER);
	    popup.getContent().add(vbox);
	    popup.show(owner, owner.getX(), owner.getY()+owner.getHeight());
	}

	public static void main(String[] args) {
		launch(args);
	}
}