If you would like to disable word-press updates please follow the below process:

Open your current selected Theme folder.
Open under this folder function.php and put the below code at the end of this file

//To Disable Theme Updates # WP Version 3.0+
remove_action( 'load-update-core.php', 'wp_update_themes' );
add_filter( 'pre_site_transient_update_themes', create_function( '$a', "return null;" ) );
wp_clear_scheduled_hook( 'wp_update_themes' );

//To Disable Plugin Updates # WP Version 3.0+
remove_action( 'load-update-core.php', 'wp_update_plugins' );
add_filter( 'pre_site_transient_update_plugins', create_function( '$a', "return null;" ) );
wp_clear_scheduled_hook( 'wp_update_plugins' );

//To Diasable Core Updates # WP Version 3.0+
add_filter( 'pre_site_transient_update_core', create_function( '$a', "return null;" ) );
wp_clear_scheduled_hook( 'wp_version_check' );