Spock should support any language running on Groovy, at least to some extent.
Might be that the integrated mocking and stubbing is sub-optimal, meaning you need to call them like you would from Java code or something like that.
Yes, you write them in Groovy.
If someone is keen enough he can probably implement a Kotlin front-end though now with the IR compiler and transformations you can do.
But the core Spock team is small and noone has time for that as it probably is a quite big task.