In Flutter, how do I pass data into a Stateless Widget?

Yes you can do this simply by passing the info to the constructor. Something like this:

 class MyApp extends StatelessWidget {
  MyApp(this.yourData);

  final int yourData;
  @override
  Widget build(BuildContext context) {
    return new MaterialApp(
      title: 'App Title',
      theme: new ThemeData(
        primarySwatch: Colors.green,
      ),
      home: new MainBody(yourData),
    );
  }
}


class MainBody extends StatelessWidget {
  MainBody(this.yourData);

  final int yourData;

  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      body: new Padding(
        padding: const EdgeInsets.only(left: 20.0, right: 20.0),
        child: new Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            new TimeCheck(yourData),
          ],
        ),
      ),
    );
  }
}