We have
$$k(k + 1)(k + 2)(k + 3) = (k^2 + 3k)(k^2 + 3k + 2) = (k^2 + 3k + 1)^2 - 1.$$
That means that $n(n + 1) + 1 = n^2 + n + 1$ has to be a perfect square, but that is impossible since
$$n^2 < n^2 + n + 1 < n^2 + 2n + 1 = (n + 1)^2$$
i.e. $n^2 + n + 1$ is between two consecutive squares.
Therefore, such positive integers $k$ and $n$ do not exist.