Change default header image in twenty fourteen child theme

Yes, there is possible.

Inside the functions.php add the follow source.

add_action( 'after_setup_theme', 'twentyfourteen_default_header' );
/**
 * Add Default Custom Header Image To Twenty Fourteen Theme
 * 
 * @return void
 */
function twentyfourteen_default_header() {

    add_theme_support(
        'custom-header',
        apply_filters(
            'twentyfourteen_custom_header_args',
            array(
                'default-text-color' => 'fff',
                'default-image' => get_stylesheet_directory_uri() . '/images/default_header.jpg',
            )
        )
    );
}

Now was the set the default image, from the path. The rest is a little bid css magic to set to your requirements. Alternative you can set also a image via CSS, but is not so easy to change for the theme users. With this function you set a default and the custom possibilities like Customizer or via back end is also possible.