Second Derivative of Lipchitz Concave Curve is infinite at only finite points
The answer is no. Consider the function
$$
u(x) = \sum_{j=1}^\infty \frac{-2^{-j}}{\sqrt{|x - 2^{-j}|}}
$$
which clearly is integrable, negative, and has poles at $2^{-j}$, i.e. at countably many points in $[0,1]$. Then set
$$
Q_0(x) = \int_0^x \int_0^t u(s) \, ds \, dt, \quad Q(x) = Q_0(x) - Q_0(1)x \, .
$$
Then $Q'' = u \le 0$ and $Q'$ is continuous, hence $Q$ is even differentiable everywhere.