Starting from the conservation of mass:
$$ \dot m_{1}=\dot m_{2} $$
This translates to
$$ \rho_{1} S_{1} V_{1}=\rho_{2} S_{2} V_{2} $$
Assuming incompressible flow, thus $\rho_{1}=\rho_{2}$
gives:
$$S_{1} V_{1}= S_{2} V_{2} $$
With $S_{1} V_{1} = Q_{1}$ , the formula you are using.
This formula follows directly from the mass balance, with only the assumption of incompressible flow. There is no assumption on turbulent or laminar flow, thus this equation holds for both flow types.
With the balance given above you can calculate the speed in at location 2 for the given value of $V_1$ and the ratio of Areas, just as you did. There is no need to account for the flow type.
However, it should be noted here that these are average speeds. If you want to go into further detail, you could include friction forces in the pipe. These friction forces depend on the flow type, and determine the shape in velocity profile, and the resulting velocity.
However, this is significantly more difficult to do than just solving a couple equations.
I know there are some rules of thumb to estimate losses in pipes, but you have to check if their assumptions are valid for your case.
Perhaps you can have a look at Pipe Flow Fluid Mechanics Course
The transition from Hagen-Poiseuille to Darcy-Weisbach behaviour happens when the flow regime switches from laminar to turbulent. Note that the DW equation has a fudge factor
that describes the energy loss due to turbulence. This friction factor varies with flow rate.
For an increase in pressure to produce no increase in flow the friction factor would have to go to infinity at some flow rate. As far as I know this doesn't happen so an increase in pressure will always produce some increase in flow rate, though that increase will be proportionally smaller as the flow rate increases.
Best Answer
Find the velocity using bernoulli's equation
Pressure + G.P.E per unit volume + K.E per unit volume = constant
and apply the equation for flow rate.
Flow rate = Area x Velocity