In this paper, the strong and doubly metric dimensions of Johnson and Kneser graphs are considered. The exact value of the strong metric dimension of Johnson graph $J_{n,k}$ is obtained using the well-known results from the literature. The strong metric dimension of Kneser graph $K_{n,k}$ has been obtained for $n \ge 3k-1$. Finally, it has been shown that the doubly metric dimensions of Johnson graph $J_{n,2}$ and Kneser graph $K_{n,2}$ are both $\lceil \frac{2n}{3} \rceil$.