As already mentioned, the conserved quantity is A itself.

I'll elaborate a little to see the full analogy between the classical and quantum mechanical case.

Let's assume that the lagrangian depends on the coordinates $q_i\in R^n$ and their first derivatives. The classical hamiltonian $H(p,q)$ which is a function on the phase space $T^{*}R^n$ can be obtained through the Legendre transform.

Now if a transformation preserves the Lagrangian, you will see that there exists a function $A(p,q)$ on $T^{*}R^n$ whose canonical Poisson bracket with the Hamiltonian vanishes: $\{A(p,q), H(p,q)\} = 0$. When you implement this transformation on the coordinates, you get the transformation law you started with: $\{A(p,q),q_i\} = K_i(q)$,

Thus, what Noether theorem really does is to allow you compute a function $A(p,q)$ which canonically generates the transformation you started with. This function is conserved under the classical evolution since its Poisson brackets with the hamiltonian vanish.

As already mentioned, the conserved quantity is A itself.

I'll elaborate a little to see the full analogy between the classical and quantum mechanical case.

Let's assume that the lagrangian depends on the coordinates $q_i\in R^n$ and their first derivatives. The classical hamiltonian $H(p,q)$ which is a function on the phase space $T^{*}R^n$ can be obtained through the Legendre transform.

Now if a transformation preserves the Lagrangian, you will see that there exists a function $A(p,q)$ on $T^{*}R^n$ whose canonical Poisson bracket with the Hamiltonian vanishes: $\{A(p,q), H(p,q)\} = 0$. When you implement this transformation on the coordinates, you get the transformation law you started with: $\{A(p,q),q_i\} = K_i(q)$,

Thus, what Noether theorem really does is to allow you compute a function $A(p,q)$ which canonically generates the transformation you started with. This function is conserved under the classical evolution since its Poisson brackets with the hamiltonian vanish.