Can a 12-24Vdc, 2A, 4 Channel light controller for strip lights be used to control (5) CREE 3.1V RGBW LED Emitters in Series?
I think that, yes, you can use that controller for your LEDs.  Note that each Cree LED has 8 terminals.  You will need to wire each of the Cree stars in series with the one next to it, giving you 4 series strings with 5 LEDs in each string.  This will give you a voltage drop somewhere near 16V for each of the colors, possibly excepting Red.  Some Red LEDs run with 1.7 - 2.1V drop, some run with a 3.2V drop similar to Green, Blue, White.  You will have to measure to find out for sure.

Do note that you need to provide current limit for each of your strings.  The distributed LED strip that you show in your picture has series resistors for each group of LEDs.  You can use active current limit or simple resistors.  The advantage of the active current limiters is that the current remains the same even if the voltage varies, so long as the voltage remains above the dropout voltage of the current limit plus the forward voltage of the LED string.

Resistors are simpler and cheaper but the current varies as the supply voltage changes.  Note that if your supply voltage is fairly close to the voltage drop in the LED string, small changes in supply voltage will result in large changes in LED current.

Regardless of which current limit technique you use, realize that it's going to get hot.  Assuming 1 Amp forward current per string running from a 24V supply rail, each current limit stage is going to be dropping about 8V at 1 Amp, which is about 8 watts.