Yes — cucumber is mostly water: safe, hydrating, and low calorie.
What it means for conures
There's nothing specific to a conure that changes the verdict on cucumber — it's treated the same as for parrots generally. A conure can handle a moderate portion.
The full picture on cucumber
Cucumber is about 96% water with mild amounts of vitamin K and potassium. It won't anchor a diet, but it's a refreshing, completely safe vegetable that suits overweight birds and hot days perfectly. Peel, seeds, and flesh are all fine.