ΔG° will differ depending on what you consider your standard concentrations (1M or 1 atm).
Quite right, and therefore I said, in the other thread I referred to, that the standard state must be that used in the definition of ΔG°.
Because the standard pressure used for Qp and the standard concentration for Qc are not equivalent,
But they must be, for precisely that reason.
Customarily, the standard state for a solution is 1M concentration, and the standard state for a gas is 1 atm pressure. This is what you will typically find for quoted values of standard thermodynamic functions.
Assuming we are talking about gases (otherwise the question of Q
c vs. Q
p will not arise), the standard state is 1 atm pressure or equivalently 1/24 M concentration. It is not "1 atm if we're talking pressure but 1 M if we're talking concentration" - these would have, as you say different G°.
In that case the normalised pressure and concentration are the same - if P
A, say, is 2 atm, then the pressure of A is twice the standard pressure, and its concentration is twice the standard concentration. Either way, the number 2 goes into the expression for Q.