I am trying to trigger an ADC conversion through TA0 CCTL1 Output in my MSP430, but nothing is happening
Table 6-12 in the datasheet also shows how the ADC trigger input is connected to the timer:



And this makes clear that the "timer output" is the same signal that would be output to a GPIO pin. In other words, this has nothing to do with the interrupt itself, but you have to configure the output mode to generate a rising edge at the desired time. The default output mode (0) never changes the output; you want something like "Toggle" or "Set/Reset", depending on the frequency: