The WordPress application (ie. from WordPress.org) does offer some default shortcodes that you can reference.
[contact] is not one of them. You seem to be referring to the Shortcodes offered by WordPress.com however these shortcodes are peculiar to sites hosted on WordPress.com – they will not work in your WordPress application.
To call other shortcodes you will need to choose a plugin that includes contact shortcodes, or write your own, as described in the WordPress Shortcode API.