I’m using a similar set up as yours- a ceiling fan with a built-in lights and RF remote control. When you tell Google Home to turn on fan, it will only turn on the “on” button on your IHC/Broadlink app as you have screenshot. Google Home can only turn on or turn off the fan, it is unable to turn on your fan light.
So if you want to turn on the lights, you have to create a scene like another brother has explained and then crease a new routine under Google Home.
Alternatively, you can create another lamp on IHC/Broadlink app, and set it to learn the RF for on and off for your fan lights, then place it on whichever room you want on the Google Home app, say for example, living room. So when you say, turn on living room lights, it will turn on that particular lamp. But do take note- if you have multiple lamps placed in the living room, Google Home will turn on all the lamps in that particular room.