Pareto efficient quantity of public good (quasi-linear preferences)
I don't think it is true in a standard public good economy the question is referring to. Consider the following counterexample:

Suppose $I  = \{1,2\}$ and utility of the individual $i$ depends on his consumption of public good $(G)$ and private good $x_i$: $u_1(G, x_1) = 2\sqrt{G} + x_1$ and $u_2(G, x_2) = 2\sqrt{G} + x_2$, 

Also, the CRS technology used for production of public good uses private good as input:
$G = f(x_0) = x_0$. 

If the society has only 4 units of private good in the beginning, then the set of feasible allocations can be written as

$\{(G, x_1, x_2)\in\mathbb{R}^3_+: G+x_1+x_2 = 4\}$.

Notice that the allocation $a_1 = (G, x_1, x_2) = (1,3,0)$ is Pareto efficient, but does not maximize the sum of utilities. The reason is that allocation $a_2 = (G, x_1, x_2) = (4,0,0)$ yields the higher sum.

$\color{blue}{u_1(1,3) + u_2(1,0)} = 5+2 =7 \color{blue}{&lt;} 8 = 4 + 4 = \color{blue}{u_1(4,0) + u_2(4,0)}$.