There's an accessory port socket underneath the white extruder cover that's empty. In theory, the M42 P15 S0/S1 commands should be toggling that on/off, but in my testing the power remained on constantly. Hopefully just a software bug.
So if I'm reading your laser's specs correctly, it needs to be powered by 12V. That means you would need an external power supply, since the Cetus provides 5V on the accessory ports (besides which, I don't think those ports could source enough amps to power the laser anyway).
On the bright side, the accessory port does supply 5V, which according to your spec above is what the TTL control of your laser needs, so that sounds like it would be perfect.
As I've mentioned before, we'll have to wait on Cetus to disclose what's happening with turning on/off that pin, though, since it doesn't seem to work with the command mentioned in the update I've linked above.
As for the actual connection, I think (but am not positive), that it's a JST-EH connector. But be careful, there's some places selling what they call JST-EH which is clearly not, so look at it before buying. I've been able to repurpose an R/C receiver servo plug to fit, though the pin-out is different.
The connector I'm talking about is under the white cover on top of your extruder. The one on the right side in the attached image.
The Cetus (and its UP siblings) don't natively speak gcode, which is what S3D outputs. So you'll need to generate your gcode file w/ S3D and then load that gcode file into Cetus Studio for printing. There's a thread talking about S3D and another talking about Cura on these forums.
In particular the parts about the coordinate system, offsets, etc.
Please also review the gcode file before running it, especially the first few lines of movements, to make sure you've got the coordinate system & nozzle offset correct.
This is very little information to give an answer, which model of laser engraver you have, which brand. There are characteristics of absolutely different types of such devices Perhaps you will find something similar to yours and will be able to understand how to connect it. Look, they are all completely different.
I’m planning to do some similar upgrades as well and I was wondering if you could share what kind of laser you use, how you connected it to the printer, how you was able to turn the laser on and of using TTL. Maybe some G-code.