Convex + Monotone =? Convex

The function $f(x)=0$ is a convex function. Thus, you would require that every monotone increasing function is convex.

$g(x)=2x+\sin x$ is strictly increasing, $f(x)=\frac15x^2$ is strictly convex. Yet, $f''(x)+g''(x)=\frac25-\sin x$, so $f+g$ is not convex.

For another example, which visibly fails to be convex and is in fact concave everywhere, add the strictly convex function $f(x) = e^{-x}$ and the strictly increasing function $g(x) = -2 e^{-x}$ to get $f(x) + g(x) = -e^{-x}$.