If you have registered your birthday in your phone's default contact app, you can add your birthday to Timeblocks by linking the contact app.
Timeblocks Menu - Connect - Contacts - Allow Permissions.
The color of the linked birthday can be adjusted by clicking the gear-shaped setting button on the right side of the Time Blocks menu - Categories - Contacts to adjust the category color.
If you want to change the birthday date or the birthday person's name, you must do so in the Contacts app, not the Timeblocks app.