How can render a Block on a Contact form Page
Following the logic exactly as you have laid it out, you could test for 

if (arg(0) == 'contact') 


to output the block on the contact page. However, is there a reason you're not using the block admin page to place this block instead? Other options here that might be easier to maintain would be:


place the block in the region you want using the Context module
copy the page.tpl.php file from your theme to page--contact.tpl.php and add your block content there