Suppose each plate of a parallel plate capacitor carries a charge q...

Let the initial charge on each plate of the capacitor be $q$.

When a battery is connected across the capacitor, it transfers charge from one plate to the other. If the battery transfers a charge $Q$, it delivers $+Q$ to one plate and $-Q$ to the other. Thus, the battery delivers equal and opposite charges to the two plates.

The final charges on the two plates become $q_1 = q + Q$ and $q_2 = q - Q$. Since $q \neq 0$, the total charges on the two plates are not equal and opposite.

The charge on the outer surfaces of the plates is always half the sum of the total charges on the two plates. Therefore, the charge on each outer surface is $\dfrac{q_1 + q_2}{2} = \dfrac{q + Q + q - Q}{2} = q$. Thus, the outer surfaces carry equal charges.

The charge on the inner (facing) surfaces of the plates is the difference between the total charge on the plate and the charge on its outer surface. For the first plate, it is $q + Q - q = Q$. For the second plate, it is $q - Q - q = -Q$. Thus, the facing surfaces carry equal and opposite charges.