If your LCD display is the conventional parallel interface type, there won't be sufficient I/O available on the QT4 device for simple direct connection of the display - as a minimum you would likely need the QY4 device.
With 4-bit parallel (nybble) interface, you will need a minimum of 6 lines from the device, all output capable. With 8-bit parallel interface a minimum of 10 lines are needed. The code is somewhat more complex for the 4-bit mode.
The number of pins required could potentially be reduced with an intervening shift register device between the MCU and the display - requiring a more complex circuit configuration and again, more complex code.
a pcf8574 with a software I2C interface will do the job using two I/O Pins.
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