$k$-clique in $k$-partite graph
This should be indeed NP-hard. And the construction is very similar to one that already worked for a similar question: Many-one reduction from inequality problem to equality problem.

From a graph $G=(V,E)$ and $k$ construct the graph $G'$ with vertex set $V \times \{1,\dots,k\}$ and edges between $(v,i)$ and $(w,j)$ if $(i \ne j)$ and $(v \ne w)$ and $(v,w) \in E$. It is easy to see that $G'$ is $k$-partite and it has a $k$-clique if and only if $G$ has.