Cournot Model Monopoly example
The example you gave is correct.

What you are missing is that you took the derivative of the demand function and not of the revenue function. What you should have done is:

$Revenue = R = Price*Quantity = (-Q + 120) * Q = -Q^2 + 120Q$

$Marginal \; Revenue = MR=\frac{dR}{dQ} = \frac{d(P*Q)}{dQ} = 120 - 2Q$

What you did was:

$MR = \frac{dP}{dQ}=-1$.

Hence, you took the derivative of the Price and not of Revenue to get marginal revenue, which is wrong.