"Serial adapter" is insufficiently precise.
LCD adapters have been made (and are popular) with both UART and I2C "Serial" connections (and SPI, too.)
If you have an I2C adapter, there are many libraries that will let you use the LCD.
If it's a UART adapter, you probably need to set up a softwareSerial and refer to the particular adapater's documentation for "commands" (AFAIK, UART adapters are less "standardized" than I2C adapters. They were popular back in the "BASIC Stamp" days, when you didn't have I2C, but could send UART data on any pin with a simple "SEROUT" command.)