编辑-新答案：

看起来像谷歌添加了适当的API调用：NotificationManagerCompat.from(context).areNotificationsEnabled()

旧答案：

对于正在查看此问题的任何人，请注意NotificationListenerService与“显示通知”不同。这两者是不同的东西！如果应用可以访问NotificationListenerService并不意味着会显示其通知，反之亦然。为了检查用户是否阻止了来自您的应用的通知，您可以使用反射：

public class NotificationsUtils {

private static final String CHECK_OP_NO_THROW = "checkOpNoThrow";
private static final String OP_POST_NOTIFICATION = "OP_POST_NOTIFICATION";

public static boolean isNotificationEnabled() {

    AppOpsManager mAppOps = (AppOpsManager) GlobalContext.getContext().getSystemService(Context.APP_OPS_SERVICE);

    ApplicationInfo appInfo = GlobalContext.getContext().getApplicationInfo();

    String pkg = GlobalContext.getContext().getApplicationContext().getPackageName();

    int uid = appInfo.uid;

    Class appOpsClass = null; /* Context.APP_OPS_MANAGER */

    try {

        appOpsClass = Class.forName(AppOpsManager.class.getName());

        Method checkOpNoThrowMethod = appOpsClass.getMethod(CHECK_OP_NO_THROW, Integer.TYPE, Integer.TYPE, String.class);

        Field opPostNotificationValue = appOpsClass.getDeclaredField(OP_POST_NOTIFICATION);
        int value = (int)opPostNotificationValue.get(Integer.class);

        return ((int)checkOpNoThrowMethod.invoke(mAppOps,value, uid, pkg) == AppOpsManager.MODE_ALLOWED);

    } catch (ClassNotFoundException e) {
        e.printStackTrace();
    } catch (NoSuchMethodException e) {
        e.printStackTrace();
    } catch (NoSuchFieldException e) {
        e.printStackTrace();
    } catch (InvocationTargetException e) {
        e.printStackTrace();
    } catch (IllegalAccessException e) {
        e.printStackTrace();
    }
    return false;
}
}

来源：https://code.google.com/p/android/issues/detail?id=38482

NotificationManagerCompat.from(context).areNotificationsEnabled()

似乎是个好办法。

以下是此问题的更完整答案

fun areNotificationsEnabled(context: Context, channelId: String = "fcm_fallback_notification_channel"): Boolean {
    // check if global notification switch is ON or OFF
    if (NotificationManagerCompat.from(context).areNotificationsEnabled())
        // if its ON then we need to check for individual channels in OREO
        return if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.O) {
            val manager = context.getSystemService(Context.NOTIFICATION_SERVICE) as NotificationManager
            val channel = manager.getNotificationChannel(channelId)
            return channel?.importance != NotificationManager.IMPORTANCE_NONE
        } else {
            // if this less then OREO it means that notifications are enabled
            true
        }
    // if this is less then OREO it means that notifications are disabled
    return false
}

试试这个：
Is there a way an app can check if it is allowed to access notifications?
这里有更多的解释：
使用NotificationListenerService检查对通知的访问

if(NotificationManagerCompat.from(context).areNotificationsEnabled())
{
 //Do your Work
}
else
{
     //Ask for permission
}

public class MainActivity extends AppCompatActivity {

   @Override
   protected void onCreate(Bundle savedInstanceState) {
       super.onCreate(savedInstanceState);
       setContentView(R.layout.activity_main);

//Main Code Below-
//NotificationManagerCompat.from(context).areNotificationsEnabled()
       if(NotificationManagerCompat.from(this).areNotificationsEnabled())
       {
           Toast.makeText(this, "Notification is on for this Application", Toast.LENGTH_SHORT).show();
       }
       else
       {
           Toast.makeText(this, "Notification is off for this Application", Toast.LENGTH_SHORT).show();
       }
   }

}

2023-03-18：Android 13（SDK 33 TIRAMISU）引入了新的运行时权限（android.permission.POST_NOTIFICATIONS），允许应用发布通知（在Android 13之前默认允许）。
如果您的应用的目标SDK低于33并且在Android 13设备上运行，则用户首次打开您的应用时，系统将自动请求此权限，然后由用户决定是否允许您的应用发布通知。
如果您的应用面向SDK 33且正在运行Android 13设备，则默认情况下发布通知为不允许，除非您明确请求且用户允许。
请注意，用户可能允许您的应用发布通知，但同时禁用特定的通知通道，因此仅测试是否授予权限是不够的，还需要测试是否启用通知通道。
下面是一个示例帮助函数（在Kotlin中），用于检查是否允许通知：

fun isPermissionGranted(context: Context, permission: String): Boolean =
    ActivityCompat.checkSelfPermission(context, permission) == PackageManager.PERMISSION_GRANTED

fun isNotificationAllowed(context: Context, channelId: String? = null): Boolean {
    if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.TIRAMISU) {
        if (!isPermissionGranted(Manifest.permission.POST_NOTIFICATIONS)) return false
    }
    if (!NotificationManagerCompat.from(context).areNotificationsEnabled()) return false

    if (channelId == null) return true
    runCatching {
        val mgr = context.getSystemService(Context.NOTIFICATION_SERVICE) as NotificationManager
        val channel = mgr.getNotificationChannel(channelId)
        return channel.importance != NotificationManager.IMPORTANCE_NONE
    }
    return false
}

试试这个：

if (Settings.Secure.getString(getActivity().getContentResolver(), "enabled_notification_listeners").contains(getActivity().getPackageName())) {
    // Notification access service already enabled
    Toast.makeText(getActivity(),"notification enabled",Toast.LENGTH_LONG).show();
} else {
    Intent intent = new Intent();
    intent.setClassName("com.android.settings", "com.android.settings.Settings$AppNotificationSettingsActivity");
    intent.putExtra("app_package", getPackageName());
    intent.putExtra("app_uid", getApplicationInfo().uid);
    startActivity(intent);
}