Budgies must be one of the most stubborn birds I have ever came across when it comes to diet!! So you do have posed one of the hardest questions.
Our little Csilli the budgie will eat only her greens - lettuce, spinach, carrot leaves, very rarely some apple. No chops, no mashes, no other veggies, she never even tried it.
In 7 years I have owned her, not once she wanted to even try to eat anything else or having her converted to pellets. Forget all the tricks in the books, budgies are HARD. Especially if they are not tame/strongly bonded to you.
And there was only one way after 7 years for her to start eating pellets or try veggies - she saw my conure eating it. She ate from his bowl so she FINALLY started eating pellets, now it is part if her daily diet along with her seed mix (having budgies purely on pellet diet is bad). I did see her nibbling on veggies from Pascal's cage, but she never recognized it as the same stuff in her cage!
I even had another rescue budgie who refused to eat any other seed other than millet. He did see Csilli eating her (temporary) milletless seed mix, he was so hungry, but he refused to eat that stuff (it's different type of grass/canary seeds). He just kept tossing it out of the bowl. We were trying this for weeks, months, but no, he was SO stubborn. He saw Csilli was eating it, and he followed her, only to toss the food.
So don't feel bad if your budgie doesn't want to eat no veggies. Perhaps start with putting fresh pieces of pepper and carrot on those kabob sticks, give daily fresh on same spot. Maybe, just maybe they might give it a try, thinking of it as a toy and maybe start at least chewing on it and realizing it's food.
....or borrow a bird who eats veggies, let the budgie see them and let it steal the new bird's food xD
Good luck on your journey!!