The unit specifies that its LED outputs are of the 12 VDC 4-pin variety. Now, there are two major designs of RGB lighting devices currently used widely. One is called just plain RGB, and uses lighting devices with a 4-pin connector that provides a common +12 VDC supply line and three separate Ground lines for the three colours of LED's in the device. THIS is the type to be used with that controller you linked to.
The other common design is called Addressable RGB or ADDR RGB or ARGB. It uses a connector that looks much like the 4-pojn one for plain RGB, but is missing one of the pins. This system supplies +5 VDC and Ground common lines, and the third line is a Control Line. The system sends out addressed daya packets on the Control Line. In the lighting device, all the LED's are grouped into Nodes. Each node contains one LED each oif the three main colours plus its own control chip that listens to the Control Line and responds only to the control packet with its unique address. So both the vupply voltage and the method of control of the LED's are very different, and these two systems are incomatible and can NOT be mixed together in one system.
There's a third less-common system similar to the plain RGB one, and it includes a fourth LED colour - white - so it is called RGBW. It operates the same as a plain RGB system but requires a fifth line (to be the fourth Ground).
Unfortunately the Controller device you linked to does not show any clear specifications nor a downloadable manual. I note that its fan output ports contain 6 pins, which is NOT a common "standard" configuration. So how those ports work, and whether or not its LED ports conform to the expected "plain RGB standard" is unknown.