The fastest way to level up on Tribble currently is to just stick to the Story missions. Ignore everything else. Story missions give the best XP/time ratio. Likely this will not hold true when it is released to Holodeck however. So, don't count on that being a reliable method when F2P launches.
The fastest method I've seen involved teaming up for an Explore mission. I am not sure how my friend set up this trick as it was many months ago, and I wasn't there to help at the start. But it involved finding a "Protect the Asteroid Base" mission; there was a method we used to complete the mission and then warp out/warp in to pick it up again. Destroying 3 squads is pretty easy and we had it set on higher difficulty for even more XP. But even with this trick you are bound by the 30 minute timer on the wrapper mission. On plenty occasions I've been able to plow through three exploration missions on my own and run into the timer, so I don't really consider this an exploit or abuse. My friend leveled to Lt General on his KDF during the first Q Bonus Weekend, though it was damn boring.
Besides that, the quickest I've seen involves Foundry exploits and that obviously doesn't count.