An app which has BottomNavigationBar displaying badge count also has app icon.

bottom_navigation_bar_with_badge.dart

import 'package:flutter/material.dart';

/// Add dependencies these four packages :)
import 'package:flutter_hooks/flutter_hooks.dart';
import 'package:hooks_riverpod/hooks_riverpod.dart';
import 'package:flutter_riverpod/flutter_riverpod.dart';
import 'package:flutter_app_badger/flutter_app_badger.dart';

/// Define providers for holding badge number.
final _homeBadgeProvider = StateProvider<int>(
  (ref) => 0,
);
final _notificationBadgeProvider = StateProvider<int>(
  (ref) => 12,
);
final _myPageBadgeProvider = StateProvider<int>(
  (ref) => 20,
);

/// Watch all providers for app icon.
final _appBadgeProvider = Provider<int>(
  (ref) {
    final homeBadge = ref.watch(_homeBadgeProvider).state;
    final notificationBadge = ref.watch(_notificationBadgeProvider).state;
    final myPageBadge = ref.watch(_myPageBadgeProvider).state;

    return homeBadge + notificationBadge + myPageBadge;
  },
);

void main() => runApp(ProviderScope(child: App()));

class App extends StatefulWidget {
  @override
  _AppState createState() => _AppState();
}

class _AppState extends State<App> with WidgetsBindingObserver {
  @override
  Widget build(BuildContext context) => MaterialApp(home: Sample());

  /// for watching app lifecycle state, addObserver.
  @override
  void initState() {
    super.initState();
    WidgetsBinding.instance.addObserver(this);
  }

  @override
  void dispose() {
    WidgetsBinding.instance.addObserver(this);
    super.dispose();
  }

  @override
  void didChangeAppLifecycleState(AppLifecycleState state) async {
    super.didChangeAppLifecycleState(state);

    if (state != AppLifecycleState.inactive) {
      return;
    }

    if (!await FlutterAppBadger.isAppBadgeSupported()) {
      return;
    }

    FlutterAppBadger.updateBadgeCount(
      context.read(_appBadgeProvider),
    );
  }
}

/// for describing tab types.
enum TabType {
  home,
  notification,
  myPage,
}

extension TabTypeEx on TabType {
  String get title {
    switch (this) {
      case TabType.home:
        return 'ホーム';
      case TabType.notification:
        return 'お知らせ';
      case TabType.myPage:
        return 'マイページ';
    }
    throw UnimplementedError();
  }

  IconData get iconData {
    switch (this) {
      case TabType.home:
        return Icons.home;
      case TabType.notification:
        return Icons.notifications;
      case TabType.myPage:
        return Icons.people;
    }
    throw UnimplementedError();
  }

  /// choose provider for observing
  StateProvider get provider {
    switch (this) {
      case TabType.home:
        return _homeBadgeProvider;
      case TabType.notification:
        return _notificationBadgeProvider;
      case TabType.myPage:
        return _myPageBadgeProvider;
    }
    throw UnimplementedError();
  }
}

/// icon Widget for displaying in BottomNavigationBarItem
class TabItem extends HookWidget {
  const TabItem(
    this.type, {
    @required this.selected,
  });

  final TabType type;
  final bool selected;

  @override
  Widget build(BuildContext context) {
    final badgeCount = useProvider(type.provider).state;
    final showBadge = 0 < badgeCount;
    return Stack(
      overflow: Overflow.visible,
      children: [
        Icon(type.iconData),
        if (showBadge)
          Positioned(
            top: -7,
            right: -12,
            child: NotificationBadge(badgeCount: badgeCount),
          ),
      ],
    );
  }
}

/// An widget which displays a badge number with red background.
class NotificationBadge extends StatelessWidget {
  const NotificationBadge({
    @required this.badgeCount,
  });

  final int badgeCount;

  @override
  Widget build(BuildContext context) {
    return Container(
      padding: const EdgeInsets.symmetric(horizontal: 6, vertical: 4),
      decoration: BoxDecoration(
        color: Color.fromRGBO(219, 73, 58, 1.0),
        shape: BoxShape.circle,
      ),
      child: Text(
        badgeCount.toString(),
        style: const TextStyle(
          color: Colors.white,
          fontSize: 10,
          fontWeight: FontWeight.bold,
          letterSpacing: 0.75,
        ),
      ),
    );
  }
}

/// Main widget with BottomNavigationBar.
class Sample extends HookWidget {
  @override
  Widget build(BuildContext context) {
    final tabType = useState(TabType.home);
    return Scaffold(
      bottomNavigationBar: BottomNavigationBar(
        currentIndex: tabType.value.index,
        items: TabType.values
            .map(
              (tabType) => BottomNavigationBarItem(
                icon: TabItem(tabType, selected: false),
                activeIcon: TabItem(tabType, selected: true),
                label: tabType.title,
              ),
            )
            .toList(),
        onTap: (value) => tabType.value = TabType.values[value],
      ),
      body: Center(
        child: ButtonBar(
          alignment: MainAxisAlignment.center,
          children: [
            FlatButton.icon(
              onPressed: () {
                context.read(tabType.value.provider).state--;
              },
              icon: Icon(Icons.remove),
              label: Text(''),
            ),
            FlatButton.icon(
              onPressed: () {
                context.read(tabType.value.provider).state++;
              },
              icon: Icon(Icons.add),
              label: Text(''),
            ),
          ],
        ),
      ),
    );
  }
}