Google calendar only shows your contact's birthdays, but doesn't have the option to notify you. With this web application you will get reminder mails for your contact's birthdays.
This is the link to the webapp with the birthday reminder. Simply press subscribe and you will receive mails on your contact's birthdays.
You can do it yourself, by downloading the BirthdayReminder.json file and adding it to Google Drive. Then open it with Google Apps Script and publish the project as webapp. Viewing the project in Apps Script will allow you to have a look at the source code.