It seems to me that that is what we are usually only able to do. Hamiltonians are operators that act on (more abstract) Hilbert spaces. By going from classic to quantum we are using our intuition to propose $H$ in some "representation" that we can work with in the quantum theory. So I think yes, we ultimately need to go and check afterwards if this works. (We always need to do this anyway.)
For a classical negative charge the classical hamiltonian would be
$$H = T+U = \dfrac{p^2}{2\mu}-k\dfrac{Qe}{r^2},$$
where $T$ is the kinetic energy, $U$ is the potential energy and $-e$ is the charge of the electron. To turn it into a quantum hamiltonian we then transform our variables into operators $p\rightarrow\hat{p}=-i\hbar\nabla$ and $r \rightarrow \hat{r}=r$, and impose $[\hat{r}_i,\hat{p}_j] = i \hbar \delta_{ij} $, which turns $H\rightarrow \hat{H}$.
In QFT people usually start writing a classical lagrangian density $\mathcal{L}$, solving the equations of motion and then quantise. Afterwards you can calculate from this the quantum hamiltonian and check if they satisfy the quantum equation of motion $i\partial_t \hat{\phi}=[\hat{\phi}, \hat{H}]$. This works for scalar fields, Dirac fields and QED.
Though bear in mind that going from classical to quantum might not be an unambiguous task. Problems can arise when we have products of variables that turn into products of operators (that usually don't commute), where we don't know which one should go first.
