How to get current tab index in Flutter

The whole point of DefaultTabController is for it to manage tabs by itself.

If you want some custom tab management, use TabController instead.
With TabController you have access to much more informations, including the current index.

class MyTabbedPage extends StatefulWidget {
  const MyTabbedPage({Key key}) : super(key: key);
  @override
  _MyTabbedPageState createState() => new _MyTabbedPageState();
}

class _MyTabbedPageState extends State<MyTabbedPage>
    with SingleTickerProviderStateMixin {
  final List<Tab> myTabs = <Tab>[
    new Tab(text: 'LEFT'),
    new Tab(text: 'RIGHT'),
  ];

  TabController _tabController;

  @override
  void initState() {
    super.initState();
    _tabController = new TabController(vsync: this, length: myTabs.length);
  }

  @override
  void dispose() {
    _tabController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      appBar: new AppBar(
        bottom: new TabBar(
          controller: _tabController,
          tabs: myTabs,
        ),
      ),
      body: new TabBarView(
        controller: _tabController,
        children: myTabs.map((Tab tab) {
          return new Center(child: new Text(tab.text));
        }).toList(),
      ),
    );
  }
}