Can I connect a PWM pin on one Arduino to an analog input on another?
Yes, you can do this with "a little programming"  or more easily by adding two cheap components.

If you connect a say 47k resistor from PWM_Out to Analog_In and a say 100 uF capacitor from Anaolg_In to ground you will be able to read the equivalent analog value to an accuracy of around 1%. Read it N times (maybe 10 times) with a delay of say 12 ms between reads and average it for a better result. This works because the PWM period is about 1 mS so delaying for a period of multiple cycles but not an exact multiple means you sample semi randomly. 

For the sort of resolution that you want (4 levels) a MUCH smaller capacitor could be used (maybe 1 uF.) But even a 100 uF capacitor is cheap.

You could instead use PWM out (analogWrite) and a digitalRead.
Perform say 100 digital reads with say 12 ms between reads (so about 1.2 seconds total) and keep two counts - PinWasHigh and PinWasLow for (obviously) high or low conditions. The contents of PinWasHigh tells you what % of reads was high. Due to random sampling you cannot rely on this for 100 levels but determining say 10 levels should give good accuracy. The sum of PinWasHigh and PinWasLow should total 100 as a sanity check.

While use of a serial port (hardware or software) or SPI allows more data transfer per time and without much effort, this simple analog method is good enough for many applications.