I've never seen this option built-in, but there are several apps/widgets on the Playstore offering a "quick boot" as well. The term is a little misleading, as it's usually not a boot at all.
For some background information, let me first refer to my answer on Can somebody explain the boot process of an Android device?. Quick-reboot usually kills everything down to the "system server", so Zygote has to start it anew. As Zygote is the parent to all apps, basically the "Android-part" of the system is completely reloaded – while the underlying "Linux-part" keeps running.
From the users' point of view, this looks like a "quick reboot": the graphical interface (all apps, including the home screen) disappear, and then the system comes up again exactly like after a "boot". But during this process, the first 4 steps of a "real boot" (as outlined in the linked question) were skipped.
I'm not 100% sure if the "quick boot" option you are seeing there is the same, but it's pretty likely. Because, if it were a "full boot, just quicker" – then why shouldn't that be used in the first place?