It is possible to add a new task connected with a contact from its page. When you view certain contact's details, you will notice a sidebar on the right. It consists of contact information as well as issues connected with them. Just click on New issue.
You will then see a small form to fill in. Provide some details and click Add. That's it!
There is also another option to add a ticket from the contact's page. You should click on the Helpdesk tickets tab under the contact's name. Here will be the link Create helpdesk ticket. This feature is available only if both Contacts and Helpdesk modules are enabled in the project.