Assume we are given a quantum field theory described by some functional. If $J$ is a Noether current, i.e. it is associated with a symmetry of the functional and satisfies $\partial_s J^s=0$ (Noether theorem), we can always obtain a conserved quantity as $Q(x^0) = \int J^0(x^0, x^1,...,x^n)d^{n}x$. What is the converse to the previous statement? I would like to know what assumptions are necessary to build a conserved current $J$ given $Q$ swith $\dot Q = [H,Q]=0$ where $H$ is the Hamiltonian of the (quantum field) theory.

The answer to the question in your title is "no" in general. Noether current and conserved charge only go hand in hand if the symmetry that gives rise to them is a continuous symmetry, such as translation (conserved momentum) or a $U(1)$ symmetry (conserved charge). A discrete symmetry will, in general, give rise to a conserved quantity that can only take on discrete values, so it cannot "flow" and there is no current associated with that conservation law. An example is inversion symmetry, with parity as a conserved quantity. There is no current associated with parity. 

