Hello-
How old is your bird? Is he an English or American budgie? (I ask because they have different nutritional needs.) What seed does he eat? What does your vet say?
I think the most important thing is to just be patient- it takes quite some time to switch them over- sometimes several months. I have taken in a number of rescue budgies that ate only seed, and I had one, Elly Mae, that took almost four months to just start tasting the veggies, though she eventually learned to love them.
There are probably better ways, but what I have found works the best is doing a combination of things. Chopping up the veggies really, really fine (raw broccoli works best for this) and mixing it in with his seed is a good way to start. Only do this in small seed/broccoli portions right before you feed it, and take it away in a few hours though or it will spoil. Also presenting the veggies as toys works well- putting some romaine lettuce, raw broccoli or carrots, apple slices, etc, in a clip and attaching that to the cage or playgym. Mine LOVE celery leaves that have been spritzed with water and whole ears of cooked corn on the cob. I feed the veggies first thing in the morning when they are hungry.
Another thing I have done is add a
TINY bit of butter and salt to cooked veggies. Some birds will go crazy over them and start eating veggies instantly (and you then gradually remove the butter) but others will just lick the butter off, so you have to watch them. And be very careful about adding only a tiny amount of salt and butter- too much is really, really bad for them.
I have found the best thing, really, is other birds that eat veggies. Do you know anyone that has budgies that eat them? Provided both are healthy, etc, if they spend some time together peer pressure from the other bird will probably be more than enough to get yours eating too.
As for which veggies, anything that you eat is fine, barring avocado and fruit seeds or pits, except for strawberries and raspberries. But be careful or red peppers- something in them will actually change you birds feather coloring if they eat a lot of it while they moult or are growing their feathers as babies. (I discovered this the hard my with some of my albino and light blue babies... I considered selling them as "rare pink-faced budgies" but decided that that would be bad.

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