Very unlikely that it will support 360hz, but we can determine for certain. What is the ACTUAL FULL model of your laptop including any submodel? Full model or express service tag should be on the back of the laptop or somewhere in the battery compartment, usually under the battery.
So, via HDMI your laptop can support up to 144hz at 1440p or lower. It can support up to 240hz at 1440p or lower. So via mini displayport your unit can support up to 240hz on that monitor since it is a 1080p display but it cannot support 360hz at any resolution.
Frankly, I wouldn't be worried about it anyhow. At above 144hz you aren't going to be able to tell the difference anyway.