They are a set of three extremely short player-made foundry missions that can be completed in one shot by pressing a single button, hence the name "quickie". There's a set at every major port of call. You can find them in the foundry by searching for "esdq" (Earth Spacedock), "sfaq" (Starfleet Academy), "ds9q" and "kdfa" (KDF Academy).
I usually go for the SFA quickie as its very close to the history question lady and I can quickly do both and net 1,920 dilithium, 50 fleet marks and some expertise in under 2 minutes. At the same time, I can also grab doff assignments from the nearby personnel officers, killing three birds with one stone