While stranded in the desert region of the Torothan homeworld, Captain Archer promised a hungry Trip Tucker that he would have Chef prepare an unforgettable dinner and then, to keep his chief engineer conscious, asked him what he would like to eat. Besides prime rib and mashed potatoes, Trip requested that his meal also include broccoli. (ENT: "Desert Crossing")
In 2152, while waiting a neutronic storm in the ship's catwalk, Travis Mayweather, Hoshi Sato and Malcolm Reed played a game of poker, using their ration packs as bargaining chips. Among the rations being played was broccoli, but it was only included after Sato ran out of desserts. (ENT: "The Catwalk")
"Broccoli" was also a nickname that Wesley Crusher applied to Reginald Barclay. It began as a form of inside joke at the expense of Barclay due to his perceived ineptness and nervousness. When Data found out about this, he didn't understand what Barclay had to do with a vegetable. (TNG: "Hollow Pursuits")
"Good, I'll look forward to your report, Mr. Broccoli... Barclay. "