According to the documentation you can Upgrade K2 4.7 or K2 Five 5.x to K2 Five (5.6), but based on my experience I would never do a direct upgrade from 5.3 to 5.6.
Whenever I've tried similar upgrades, skipping some versions, usually something went wrong in the process and it didn't go right the first time with all markers green.
So I prefer to do the upgrade the longer way, incrementally, and sometimes I include installing cumulative updates before installing the next major version.
I definitely recommend backing up your virtual servers well with K2, and of course its databases, to be able to restore them in case of trouble, and to test beforehand. Every customer environment is different, specific in some ways, and even if you get away with the upgrade in one environment, it may fail on something in another.

If you don't feel up to it, you can order an upgrade or assistance from Nintex Professional Services.

Good luck.

Hi KimL,

I just came across this thread. We are going to upgrade our K2 Five 5.3 version to K2 Five 5.6. I was wondering if you have completed your upgrade and ran into any issues. Thank you in advance.

Regads,

Salu Joseph

Known Issue: K2 5.6 upgrade failing on the configuration service. | Community (nintex.com)