The serial port is a 2x3 pin connector (labelled JP1) that is located close to the button on the left side on the front of the mainboard. If you look at the picture below, you'll see that the layout of the row closer to the edge of the mainboard is N/C, N/C and 3.3V whereas that of the other row is RX, TX and GND. In the picture below, the red cable is 3.3V.
In order to connect to this serial port, you need a 3.3V TTL to RS-232 level shifter. The serial connector on the TS-109 is the same as that of the TS-101 which has a page with more information about the serial port (please note though that the connector has a different location on the TS-101).
Once you have connected your serial console, you can use a terminal emulation program to connect to U-Boot, the boot loader used by this device. You have to connect with 115200 baud and set the parameters to 8N1 (8 data bits, no parity, 1 stop bit). U-Boot on a separate page about U-Boot.
If your serial console doesn't work or you can see text but your typing is ignored, it might be because you have to connect RX from your connector cable to TX on the QNAP and TX with RX.