OP's eq. (1) is true for any invertible coordinate transformation -- they don't need to be canonical coordinates. The trick is to keep track of what is kept constant during the partial differentations. In 2D eq. (1) reads:

$$\begin{align} {\rm LHS}~=~&\left(\frac{\partial Q}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_q \left(\frac{\partial Q}{\partial p}\right)_p\cr
~=~& \left[\left(\frac{\partial Q}{\partial q}\right)_P + \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial Q}{\partial P}\right)_q\right] \left(\frac{\partial P}{\partial p}\right)_q - \left(\frac{\partial P}{\partial q}\right)_p \left(\frac{\partial P}{\partial p}\right)_q \left(\frac{\partial Q}{\partial P}\right)_q \cr
~=~& \left(\frac{\partial Q}{\partial q}\right)_P\left(\frac{\partial P}{\partial p}\right)_q
~=~\left(\frac{\partial Q}{\partial q}\right)_P / \left(\frac{\partial p}{\partial P}\right)_q
~=~{\rm RHS},\end{align}$$
where we used the multi-variable [chain rule](https://en.wikipedia.org/wiki/Chain_rule) twice.