<TextBox Width="50" Text="{Binding ElementName=dataGrid1, Path=SelectedIndex, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}"></TextBox>

[ValueConversion(typeof(int), typeof(string))]
class DataGridSelectedIndexToStringConverter: IValueConverter
{
    public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
    {
        int index = (int)value;
        return (++index).ToString();
    }

    public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
    {
        string strValue = value as string;
        int index;
        if (int.TryParse(strValue, out index))
        {
            return index;
        }
        else
        {
            return DependencyProperty.UnsetValue;
        }
    }
}

<my:DataGridSelectedIndexToStringConverter x:Key="DataGridSelectedIndexToStringConverter" />

  <TextBox Width="50" Text="{Binding ElementName=dataGrid1, Path=SelectedIndex, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged, Converter={StaticResource DataGridSelectedIndexToStringConverter}}"></TextBox>