Well, you possibly could use them, but you wouldn't be able to get TTL, HSS, remote power control, group control, etc. with any of it. You'd only be able to fire the flash remotely. You'd have to use it in M, below sync speed, and make any settings changes directly on the flash.
The YN-600EX-RT whether used with a Canon RT-compatible trigger (e.g., YN-E3-RT) or with the Godox X1T/X1R transmitter/receiver set will only give you TTL/HSS with a Canon camera body. The X1R receiver does not do cross-brand TTL/HSS communication the way a built-in transceiver in a Godox TTL/HSS capable speedlight would. And it's also possible the X1T-C transmitter would not ground properly on a Sony multi-interface hotshoe and work at all as a manual transmitter to simply fire the flash remotely.
Your best bet would probably be to ditch everything altogether, and get Sony-compatible flash radio gear (Godox, Westcott FJ, Yongnuo, whatever) both flash and transmitter that work with the Sony multi-interface hotshoe. If you need to maintain TTL/HSS compatibility with Canon gear as well, the Westcott FJ (aka Jinbei RT) gear would actually work in TTL/HSS on both types of hotshoes (you wouldn't need to purchase duplicates), but you would need an adapter for the Sony hotshoe.
With Godox, you would need separate transmitter units, but you could use the same speedlight for off-camera work. But the speedlight could only do TTL/HSS on-camera for one system or the other. Whatever goes directly on the hotshoe of the camera has to match the camera's brand (e.g., an X1T-S for Sony and an X1T-C for Canon).
I use the Godox trigger on Canon because of the hotshoe on top; I can't find a similar trigger on Yongnuo.
The closest similar trigger Yongnuo makes would be the YN622C transceivers, which are a PITA to use as transmitters (no display; you tend to have to do everything via the camera menus), and are in a completely different incompatible radio system to the Canon RT cloned system. With Canon gear, you would ideally pair the YN-600EX-RT II with a YN-E3-RT for off-camera use.
Do I only need to upgrade the transmit trigger or also the receiver? The Canon receiver will not work with the Sony Godox transmit trigger, right?
As I mentioned above, the X1R will not work for TTL/HSS cross-brand for a for-Sony Godox transmitter, so will become a manual trigger (no TTL, no HSS, no remote power control). All it can do is fire the flash remotely. But yes, it will do the fire communication with a Godox for-Sony transmitter, like an XPro-S.
But as I said above, the best option would be to replace the triggers and flash you have with something like an TT685 II-S and an XPro-S. You could then use the TT685 II-S both on-camera and off-camera with TTL/HSS via the Xpro-S on your Sony gear, and off-camera with the X1T-C on your Canon gear, and you wouldn't need to attach any receiver to its foot.