Is the convexity of production sets necessary for the welfare theorems? I have read that the convexity of production sets (for instance non-increasing returns to scale) is not a necessary assumption for the first welfare theorem but it is for the second welfare theorem. Maybe I got things wrong and what the author actually tried to say is that convexity is not necessary for the efficiency of the competitive equilibrium but rather for the existence of the competitive equilibrium.

Could you please elaborate where the assumption of convex production sets is needed in the first and second welfare theorem?
Convexity of the production set is indeed not needed for the proof of the first welfare theorem but for the proof of the second welfare theorem. It is not a necessary condition though. 

It is possible to interpret this as an existence issue. The first welfare theorem is about all competitive equilibria and holds trivially if there are none. The second welfare theorem, on the other hand, states that for a given Pareto efficient allocation, there is a price system and redistribution of endowments with respect to which it is a (quasi-)equilibrium.

The standard proof of the second welfare theorem uses a result of Minkowski on the separation of nonoverlapping convex sets by a hyperplane, but it is possible to prove a version of the second welfare theorem as a corollary to an existence result by a nice argument due to Maskin and Roberts. The argument is quite easy in the case of an exchange economy: Take a Pareto efficient allocation as the endowment distribution. If a competitive equilibrium exists from these endowments, everyone will end up with something at least as good as their endowment. Since the endowment distribution was Pareto efficient, nobody can end up with something better. So everyone must be indifferent between their demanded commodity bundle and their endowment, so they might as well just demand just their endowment. The argument generalizes to economies with production.