Quick answer: No, the PS4 does not offer native Discord support. It also does not offer Rich Presence integration. There are workarounds to both shortcomings, but they are far from being plug and play or, arguably, worth the effort.
Despite the PS4’s mature (and updated) party chat functionality, its utility is still trounced by comparable services on the market. Discord, the de facto option for PC players, is one such service that offers an unparalleled feature set, better stability, and superior VoIP quality.
While the official party chat functionality may offer convenience, many Sony players have taken to using the official Discord app in order to organize play sessions with randoms and friends alike. The improved call quality and ease of use features are just a couple reasons why it’s become used by console players over the past few years.