* Correct: My child attends the Bilingual Kindergarten program at Lincoln Elementary.
* Correct: She graduated from the Bilingual Kindergarten at Maplewood School.
If you're referring to the concept of bilingual kindergarten in general, you don't capitalize it:
* Correct: The school offers a bilingual kindergarten program.
* Correct: Bilingual kindergarten is beneficial for early language development.
In short, capitalization depends on whether it's a specific named program or a general concept.