We have some fleets on STO, although some of them make events on their own, there isnt a good interaction between fleets. There isnt something for fleets to compete for, there isnt events/mission made for them.
Now its just a bound of friends, doing STFs, over and over, but we can improve this.

"Priority One Message From Starfleet Command" = Borg Invasion at Sol System

We could have once a week, prob during weekend this mission. You go to SOL System to defend it from Borgs, you can only Team-up with other fleet members and of course you need to be in a fleet to participate.
First it starts by Jupiter Station (no big deal just make another station in Sol System), you have to defend it from borgs Tactical Cubes and some Command Ships, if battle continues for 30 minutes you go to ground and defend ESD and Starfleet Academy from invading Borg Troops.
So each kills you made, counts as points for your fleet. Top 3 fleets in the end of event (if its sucessfull)will gain special rewards, according to rank.
Other fleets can convert these points to energy credits/dilithium or special itens.
We could have some cool ones, like a poker table (fleet leader will receive it), you can put at your starship and invite other players for a poker (mini game like DABO), we could have Dart game too like in DS9. Some new mini games to enjoy

How about something like "Last Man Standing"

see how long you can go, or how many Pts you get or how much damage you do.

Sure and if you die, you are kicked from this event, could work.