We've got a PoE speaker that can handle alarm inputs and alarm outputs, unfortunately I have no idea what this type of connector is called. I've look through online component catalogues and come up short. The documentation didn't even have a wiring diagram and the supplier has been less than helpful.
It's a "pluggable terminal block" or a "removable terminal block".
You need to know the pitch – spacing between pins – as a starting point. They are usually 5.0mm OR 0.2" (yes, it makes a difference) but it looks like the power block may be different than the remainder, so it might be 3.5mm.
Once you know the pitch, you need to look at pictures to find the particular type – each manufacturer uses a slightly different shape and they are not interchangeable.