They have been doing so for a long time in the Province of the Americas… During the daily Mass, they pray for all the confreres who have their birthday that day. A good initiative! After all, if we mention in our daily prayer the confreres who died that day, why not pray for those who are alive and celebrating their birthday.
You may have noticed, on the home page of the international website, the confreres who have their birthday are mentioned. If you go to the menu Society>Calendars>Birthdays, you will have the list of the confreres who celebrate their birthday during the next 7 days.
And if you want a small booklet that gives you all the birthdays of the year, to leave on the altar, for example, you can download it here, or you can find it in the menu Downloads>Various Documents. You will have to print the PDF document recto-verso turning the pages on the small side. You might have to experiment first with your particular printer.