Calabi–Yau condition Let $M$ is real $2n$-manifold with integrable complex structure. Holomorphic triviality of canonical line bundle $K$ $\longleftrightarrow$ existence of a nowhere-vanishing section in $C^{\infty}(K)$. Is it correct ? Or only one of conditions implies the other one?

No, this is not correct. The existence of a nowhere vanishing $C^\infty$ section of $K$ just means that $K$ is trivial as a smooth (or equivalently, topological) line bundle, which is weaker than being trivial as a holomorphic line bundle (for that, you would need a nowhere vanishing _holomorphic_ section).

An example of a complex manifold whose canonical bundle is topologically trivial but not holomorphically trivial is a hyperelliptic surface.