Consider the Kneser graphs $G=K(n,k)$. Is it possible to list how many even cycles, or, at the least, existence of an even cycle of a given order in $G$, such that any three consecutive vertices form a triangle?

For Odd graphs of having $n-k\ge3$, as these are triangle free, the answer would be none. But, suppose $n=7a,k=2a$, for positive $a$. Then, is it possible to say that these contain all even cycles of order $\ge12$ such that any three consecutive vertices induce a triangle? Any hints?