You do not need anything else. A 3-pin fan CAN be plugged into a 3-pin OR a 4-pin mobo fan header and it WILL work. The 4-pin fan design has NOTHING to do with lights. The 4th pin added is only for passing to the fan a speed control signal called PWM for a different control system. The speed of YOUR 3-pin fan is controlled by the Voltage supplied to it from header Pin #2.
However, if that fan always runs fast and never slows down, there is a configuration item you should check. MOST 4-pin fan headers on a mobo today can ne set to use either the older Voltage Control Mode (aka DC Mode) of control signals, OR the newer PWM signals. You should go into BIOS Setup and find the header where that fan is plugged in. Look for an item where you can choose between Voltage or DC Mode, PWM Mode, and maybe Automatic Detect Mode. Set this for Voltage (or DC) Mode for your 3-pin fan. Then ensure that the PROFOLE item is set to Standard or Normal, and not Manual or Quiet or Turbo. After making changes, use Esc back to the Main Menu, then F10 to get to the Exit Menu. There choose to SAVE and EXIT to save your settings and reboot.
NO other connection or adapter is required for the lights to work in your fan. If the fan it turning at a normal speed, the lights should be on. If they are not, then the lights are faulty. That is, of course, assuming the fan IS one with lights in it. The Deepcool Model RF120B you cited DOES had BLUE lights in its frame.