If you just want to optically control the SB-600 and SB-800 the way you did with a D7000's pop-up flash, you could purchase an SU-800 and use that on the Z6ii's hotshoe as the commander unit. You don't have to use radio triggers.
If you are going to use radio triggers, you need a transmitter unit to put on the hotshoe of the camera to act like a commander, and each speedlight will need a matching receiver unit attached to its foot. It's possible to cable a receiver to the sync port on the flash, but this connection can only communicate the "fire" (sync) signal, not additional features like TTL or HSS/FP.
So the main thing you need to make sure any radio triggers you purchase have, is Nikon iTTL support. There are a large number of radio triggers that can do this (for example, Yongnuo YN-622N, Godox X1R receivers, Pixel Kings, PocketWizard TTL trigers, RadioPoppers, etc.) Any triggers that do iTTL will have to have all four of the Nikon flash foot pins on the transmitter, and contacts on a receiver hotshoe. It's the same for other camera brands, the transmitter/receivers need to match the brand's flash feet and camera hotshoes physically if there's going to be full TTL/HSS communication.
If the transmitter only has one pin on the foot or the receivers only have one contact on a hotshoe, those triggers cannot support TTL/HSS. If a trigger is described as being "universal" for multiple systems it's likely manual only (Cactus V6II would be an exception, but it's out of production). Any triggers that have to connect via a 3.5mm, 2.5mm, or PC sync port is manual-only.
Secondly, you may want to consider whether or not its worth sticking with legacy speedlights that only work on one system of camera, when there are newer modern affordable 3rd party options that have built-in cross-brand radio remote control and better future expansion options to larger lights like studio monolights and battery-powered location strobes, like the Godox system.