This article explains how to store contacts birthdays in HubSpot
If you want to add a birthday field (property) in HubSpot, use a standard HubSpot "Date of Birth" Property.
This property is a single-line text field that is predefined by HubSpot and commonly utilized for Facebook forms. However, it can also serve as a convenient way to store the birthdays of your contacts within HubSpot.
If you want to create a custom Date property, it's also possible. Just go to Settings - Objects - Contacts - Properties - Create Property - choose "Date Picker".