This is what I understood from Klein Gordon equation :
We start from $$E^2=p^2+m^2.$$
We quantize it replacing $E \rightarrow \partial_t$, $p \rightarrow -ih\nabla$, $m \rightarrow m$
Thus, we get the Klein Gordon equation :
$$ (\Box +m^2) \Psi = 0 $$
But we can't interpret directly $\Psi$ as a wavefunction (it leads to some incoherences).
But we can interpret $\Psi$ as an operator acting on an hilbert space.
It is linked to what we call "second quantization" ? I mean, we first quantized the relation $E^2=p^2+m^2$, and we quantized the solution $\Psi$.