Sorry Raspberry Pi is not capable of this on its own. However alternative methods exist.
A possible alternative is coupling the Pi to a USB-capable Arduino like Leonardo, Due or Zero. The Pi parses the USB commands from the game controller and relays it over some other interface like SPI, I2C or UART, and the Arduino reconstruct the command exactly to the gaming console. This will introduce a lot of delay though and probably ruin your game experience. A better method is to use a real deal USB monitor, Expensive but it introduces virtually zero latency.