Let $\Omega$ be a set, e.g $\mathbb{R}^n$ for $n \in \mathbb{N}$. Let $f$ be a convex function from $\Omega$ to $\mathbb{R}$. Let $g$ be a quasi-convex function from $\Omega$ to $\mathbb{R}$.
Is $f+g$ quasi-convex? The proof is not straightforward from using the definition: Let $\lambda \in \left[0,1\right]$. Let $(x,y) \in \Omega$. Then, $\lambda (f+g)(x)+(1-\lambda)(f+g)(y) \leq max(g(x),g(y)) + \lambda f(x)+(1-\lambda)f(y)$. which does not help to conclude anything.
The statement is wrong for $\Omega = \mathbb{R}$.
Let $f(x)=-x$ and $g(x)=x-\frac{1}{2}|x|$. $f$ is obviously convex, and $g$ is monotonically increasing, and thus quasi-convex, but their sum $(f+g)(x)=-\frac{1}{2}|x|$ is obviously not quasi-convex.