The short answer
Yes — carrot is an excellent everyday vegetable for budgies, and the leafy tops are great too.
Why
Carrot is loaded with beta-carotene, which the body turns into vitamin A — essential for a budgie's eyes, skin and feathers. The green tops are even more nutritious than the root.
How to feed it
Grate it, or offer thin sticks or shavings; raw is perfect. Include the washed carrot tops when you have them.
How much & how often
Can be part of the daily fresh-food mix.