If the pump is PWM controlled, it should be working by connecting it to a fan header.

It depends on the Pump. There are a lot of AIO Pumps on the market, and only a few are compatible.
Some pumps work, some pumps need a Pull up resistor, some pumps dont work.
Please link your favorite AIO.

The Aquasuite Software can only control Aquacomputer devices. It cannot control the motherboard header.

Yes. Works. I controled over Auqareo + Aquasuite standard DDC Pump no problemos

Citation de "Netherspark"

Is Aquasuite able to control the pump on 3rd party AIOs? And how would that work - do I just connect the pump cable to a PWM fan slot on the Quadro?

As Hufeisen said, depends, but if it is a fairly standard PWM pump, then Yes.
The QUADRO has the pull-up circuitry built in, so that should not be an issue, but which AIO are you looking at?

Citation

or can it control it via the motherboard's pump header?

No

Connect your AIO pump's PWM lead to one of the four ports on the QUADRO, and connect the QUADRO's USB to a USB 2.0 port on your motherboard.
When you open the aquasuite, you click on the QUADRO tab on the left side, and then click on the 'Fan' tab under the QUADRO tab.
One of the 'Fan' ports will show your AIO pump.
I would use the 'Power preset' mode, and set the speed you want the pump to run by adjusting 'Output power'

Here is a sample of what it may look like: aio_setup.jpg (I had an open slot I used for this demo)