>If I understand correctly, wind force increases quadratically as airspeed increases

**Correct**. Anyway we speak more correctly of "aerodynamic" force.

>Therefore, all other things being equal, lift should also increase quadratically.

If everything remains exactly the same then **yes**.

Anyway this is not how an airplane deals with higher speed. The aerodynamic force depends not only on the speed but also on:

1. the fluid density
2. the dimension of the aerodynamic body 
3. the shape of the body
4. the angle with which the airspeed hits the body, more or less in a linear manner. This angle is termed "angle of attack" and called $\alpha$. Comprimibility effects (Mach > 0.3) and structural deformations modify this simple linear relationship but the main idea still holds.

***

In general the equation expressing an aerodynamic force looks something like this:

$F=½ \rho V^2 S C_F$

Where:

* $\rho$ is the fluid density - see previous point 1.
* $V$ the speed
* $S$ a surface which is used as reference area (for an airplane it is normally the wing area, for a car is the frontal surface) - see previous point 2.
* $C_F$ is the part depending on the shape of the body and $\alpha$ - see previous point 3. and 4.

If for example we consider the lift, then that equation becomes:

$L=½ \rho V^2 S C_L$

Where $C_L$ (lift coefficient) has the following shape (underlined in blue):

[![Lift, drag and moment coefficient of NACA 0012 airfoil][1]][1]

As you can see, $C_L$ increases linearly with $\alpha$ (till a certain point called stall) and can be used to change lift as needed.

Let's do a simple example: if the speed doubles, its effect on $L$ becomes four time as big. If we reduce $\alpha$ from 10° to 2° then the $C_L$ decreases from 0.8 till 0.2 i.e. four times, compensating for the increase in $V$.

[1]: https://i.sstatic.net/zCIpE.jpg