First, you always need to define a route for an action. If you don't, rails doesn't know that your action exists (even if you specify the controller and the action names in your link_to).
For that, you can simply do, in your config/routes.rb file:
get 'relationships/req'
Now, your req action has a path, relationships_req_path (responding to HTTP GET requests).
Then, if you want to call a controller action while staying on the same page, you can do:
link_to "Add as friend", relationships_req_path, remote: true
The remote: true
modifies the link behavior(it will works like an ajax call) and renders the relationships/req.js.erb file by default (which can contain nothing). This file allows use to dynamically add/modify content on the current page.