Maximize distance between k nodes in a graph
No, the problem is NP-complete.

Let $(G,k)$ be an instance of INDEPENDENT SET.  Construct $G'$ by adding a universal vertex $u$ to $G$.  The crucial observation is that the distance between two vertices in $G$ is 1 in $G'$ if and only if they are adjacent in $G$, and 2 otherwise.

Now, the optimal solution of SCATTERED SET on input $(G',k)$ is $2\binom{k}{2}$ if and only if $G$ has an independent set of size $k$.