Yesterday I asked the following question to which abx has given a positive answer.

examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

But I suddenly realized that those Fano manifolds I mentioned in the quesion above (flag manifolds $G/P$ and Fano contact manifolds) not only have trivial odd Betti numbers but have stronger properties: their Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$.

But the example abx mentioned (some smooth complete intersections) and the K3 surfaces seem not to support the above property.

So I would like to modify my original question by asking the following: does there exist a compact Kähler manifold whose first Chern class is trivial such that its Hodge numbers satisfy $h^{p,q}=0$ whenever $p\neq q$?