With the release of Fortnite Patch v4.5 today, a new feature has been revealed in the settings menu which allows you to toggle the ability to receive gifts from others:

Though no gift system is currently in place, we should expect something to arrive in the future as was later confirmed by Epic employee MrPopoTFS on reddit:

Card

No information has currently been released on how the feature will work or what items will be available for gifting.