NO. This is an older fan design called a LED Fan. It has only ONE colour of LED in the frame and there is NO adjustment of the lights. The LED's are simply connected in parallel with the fan motor so they come on whenever the fan motor is running. There is only ONE cable that goes to a fan header. As OP says, this is an older 3-pin Voltage Control fan motor design. With that system the motor speed is controlled by the Voltage supplied to it by the header, and that means that the BRIGHTNESS of the lights will be reduced when the motor is running at low speed.
OP, do NOT connect BOTH the 3-pin small fan connector AND the 4-pin Molex larger connector at the same time. You must use only ONE of these options. Disconnect the Molex and use only the mobo fan header, and ensure that the header is configured to use the older Voltage Control Mode (aka DC Mode), not PWM Mode. When that is set, if the lights still do no come on even when the fan motor is running fast, then the LED's in the fan have failed.