It's not an easy task.

Does your 1060 have VGA out? Your keyboard and mouse will have to also support the PS/2 protocol if you want to use it with the 486 (and have to have it connect somehow as a modern KVM will likely not be PS/2). USB wasn't around then and using a USB KB/M likely will not work even with an addon card.

It's why I've relegated to using a separate VGA monitor and PS/2 kb/m for my retro machines.

The age difference is just too great for this one. Your 486 PC has VGA video output and AT keyboard input, probably serial mouse. Your modern PC is probably all Display Port (and HDMI) and USB.

There are PS/2 adapters for both AT and serial ports, though you'd need to go to USB with both. VGA to HDMI adapters also exist, but in my experience, those can confuse KVM switches.

The monitor is probably the easy part, some modern monitors still have VGA input and you can switch between inputs as you need.

Lets break it down by each device

Monitor
First thing to check is max resolution supported by a KVM as VGA KVM's struggle with modern HD resolutions if this is what your main PC is running at. for example the Belkin mentioned has a max resolution of 920 x 1440 @ 75 Hz, Assuming that's not an issue
Looks like some GF 1060's have DVI, if you do you could get a DVI to VGA adapter which will allow you to then connect to a VGA KVM you will loose image quality, if it's noticeable or not depends on the person.
Another option is if the monitor has more then 1 input. Keep your main PC with its currant connection and use the monitors VGA or DVI get a VGA-DVI cable and connect this to the KVM.
This works as KVM's only need the keyboard connected that station to become live. you would need to switch between inputs using the screen itself.

Keyboard
You have KVM's like the Belkin Omniview Pro3 which support both PS2 and USB where you can buy either PS2 or USB style cables to run between the PC and KVM.
Fancy modern keyboards with loads of macro buttons, lights and the rest can be a bit tricky as they are now HID devices and rely on windows drivers and I doubt the KVM supports this. You could test with a simple USB to PS2 adapter direct to the 486 first though. This is only a somewhat recent thing so even a keyboard from a few years ago will probably work.

Mouse
This is the most tricky as they dropped backwards compatibility much earlier. Luckily this is the smallest item so its no big deal to have 2 on your desk each attached direct to the PC if needs be.