Some random ideas on how to get your bird to eat veggies.

Remember, persistence is key. It took me 2 weeks of daily trying to get my birds to eat pellets, for instance. (How I did it.) But for the birds' health, it really is worth it to get them to eat pellets and vegetables. (Remember: NO avocado or chocolate.) Generally, the younger the bird, the more success you'll have.

Why veggies? Better health, longer life, variety in the diet, etc. etc.... Read Bird Talk.

Some Random Ideas

Your mileage will vary

1. Know the basics of bird food. First of all, if you're a new bird owner, read about these common food-related mistakes first! Also, be aware these techniques work best with a friendly, hand-tame bird who thinks you are a good rolemodel, not some terrible birdie-eating monster.
2. The see-it's-not-poison-approach. Eat fruits and veggies in front of the bird. Show it how tasty the stuff is (even if it isn't). Keep doing this until the bird looks curious, if not green with envy. Then try the finger-food technique below.
3. The green paper approach. Stick leafy veggies like romaine lettuce in the cage bars (or stand some on the cage). If your bird likes to chew holes in paper, like my tiel does, then it might just mistake it for green paper. By the way, romaine lettuce is, I believe, seven times more nutritious than iceberg, and seems to be preferred by birds. You can try spinach and other dark leafy greens, too (though there is some concern about too much spinach, so don't overdo that particular veggie).
4. The fake green seeds approach. Put broccoli heads out where the bird can nibble on it (or offer some in your fingers). Broccoli heads, in particular, tend to look like a collection of green seeds; this is how I introduced the foreign concept of veggies to my 'tiel, back when he was very young. I give my birds entire broccoli stalks (with all but one of the floret sets cut off for the humans' dinner), and they sometimes rip off all the flowers and strip the stalk. You can usually tell how fresh and tasty the broccoli was by how much of it is eaten (the older and more bitter ones sometimes get no damage). If you leave the stalk on the cage-top, the birds can play with it there (and you'll probably have to go retrieve it from the floor when they push it off).
5. The finger-food technique. Offer bits of veggies in your fingers. Or, if you can do it without getting yucky human saliva on it, have one end of the food in your mouth and the other by your bird's beak, thereby proving to your paranoid bird that it's not poisonous (don't do this in civilized company, and then give the bird a clean piece if it decides the stuff is OK to eat after all). My 'tiel learned to eat green peas from my fingers. Of course, I had to puncture the pea early on because he didn't know to do that at first. (A pea, after all, is nothing more than a giant seed --- and 'tiels seem to love seeds). Now he loves to sit on the table and eat the insides out of peas --- as many as 5 (Tcsh, 10 or more!) at one sitting. The peas are probably as big as his brain....
6. The play-thing approach. Make veggies look interesting. Maybe use a knife to make a thick piece thin enough for a small beak (a big chunk of carrot would be the equivalent of giving a child a beachball-sized carrot and expecting culinary enjoyment), and give it interesting projections and things that could tempt a birdie to nibble on it and generally treat it like a toy. Maybe some of it will wind up in the bird.
7. Feed other foods, too, like plain spaghetti, bread, boiled egg (eggs should be boiled 15+ minutes, they say), and low-fat crackers. Though not veggies, encouraging your bird to be adventurous with other foods will encourage it to be adventurous with crisp green and orange things, too. My tiel just naturally seems to regard plain cooked spaghetti as some sort of intrinsic part of a cockatiel diet. Must look like a worm or something. Spaghetti dangling from fingers, set on a shoulder, sitting on a plate, or curled on top of a bird cage --- doesn't matter where.
8. Tempt with toys ... while most of these techniques work best with tame birds, this is something that I've done with my untamed budgies (parakeets). I very tightly tied small bundles of dried alfalfa together and dangled them in their cage. After a few days of comical fear and avoidance the budgies learned to chew and rip apart the alfalfa. However, I should note that dangling strings/wires can be extremely dangerous, and open loops of string/wire even more so. If anyone can think of a better way to present small bundles of alfalfa for birdie destruction, I'd love to hear about it....
9. The boring approach. You can do the standard way of adding veggies to the bird's food bowl diligently for weeks on end, too. But try the other methods as well. They're more fun and probably more productive.