The header you outlined (#8 in your posted diagram) is used to connect an optional internal case speaker to the motherboard. #15 is the onboard piezo buzzer which creates system beeps.
Now, that being said, is there a reason you want it to beep every time you restart it? If the system is properly working, sometimes it is best to leave well enough alone before you frag something up. The majority of times you have a problem, it will be covered by the diagnostic lights as described in your owners manual.
There will be an option somewhere in the system BIOS options to activate/deactivate the onboard speakers. It is possible that this option turns both the piezo buzzer and the onboard speaker header on and off. If it is deactivated, it won't hurt to turn it on if you don't actually have a case speaker plugged in to the motherboard to see if it makes a difference.