How to get or know the slug of home.php

Use is_home() for the home page (the list of newest blog posts), and is_front_page() for the front page (can be a static page too).

if( is_home() ) 
{ 
    wp_nav_menu( 
        array(
            'menu'      => 'Main Nav Menu',
            'container' => ''
        )
    );
}