I've been told by many individuals that First Contact missions are essentially broken since Season 6 and it's been left on the bottom end of the priorities list.
For the most part, you'll be able to get your first FC mission but after that, apparently, there's a timer that never resets for some people. I've been fortunate enough to get them from time to time but I'm not sure how to reproduce the conditions.
Like Datura said, the DOFF missions are great way of leveling up your Diplomacy. I attained quite a handful of Diplomats, Advisors, and Counselors and began jumping from map to map and planning Diplomatic missions. You'll find that in Defera and in the Klingon home sector, you'll be given a lot of diplomacy missions as well.
Also, the Defera Daily is an easy method to gain minor Diplomacy XP if you have time to throw away. I typically choose the 20 hour DOFF Diplomacy Missions. And from time to time, Cryptic does the commendation bonus events. That, in unison with the tips above is how I managed to reach the 4th Tier. Good luck!