SainSmart 5V 4 Channel Relay momentary switch
According to the mfg page for the SSR board you're proposing to use, it is intended to switch 75-264V AC. It's very possible that it will not correctly switch the 3 - 5 VDC for the power/reset ports of your motherboard.

The good news is that there is a 99% chance that your motherboard is just using logic inputs with a pull-up resistor to 3.3V or 5V. If so, you can probably directly connect these to your Microcontroller I/O pins.

First, use a multimeter in DC amp mode (use the lowest scale you can set) and connect the leads across the PWR switch pair and then across the RESET switch pair. When you do this, you should see the motherboard respond as if the button was pressed (the amp meter is acting as a short circuit) and observe the amount of current registering on your meter. If you see less than ~4 ma, you should be good.

Next, switch your meter to DC volts and measure again across each pair of pins and verify that it reads 5 volts or less. Also, note which pin is positive and which is negative.

You can now wire it up. Connect your micro's ground to one of the two negative pins. Connect I/O pins to each positive pin.

You'll need to program the I/O pins to "open drain" and then you can write a 0 to the pin to "press the button" and 1 to "release the button".