I am trying to trigger input pins on a controller (which I am not able to modify) with the raspberry pi. I know that the other controller's input pins are pull up, so I need to pull them down to trigger them. Will the raspberry pi pull down the controller pins?
Yes.
The Raspberry Pi GPIO will pull an external pin down to ground.
For instance the Pi has hard wired 1.8kΩ pull-ups to 3V3 fitted to GPIO 2 and 3 (pins 3 and 5) and can pull them down to ground when GPIO 2 and 3 are set as a low output.