HP Z600 Workstation, with windows 10 Pro, will a ATI RADEON HD 4870 card work in this computer??
Yes it should work as long as it will fit inside the computer case except possibly BIOS may have to be in CSM Mode or Legacy mode. Not sure if the GPU card is compatible with UEFI mode.
But why do you want to install such an Out-dated and unsupported GPU Card that doesn't even have any AMD Windows 10 drivers at AMD Download page except what is installed by Windows Update? https://www.amd.com/en/support/graphics/amd-radeon-hd/ati-radeon-hd-4000-series/ati-radeon-hd-4870
Plus it only support DX10 and nothing higher.
Plus it doesn't support Vulkan API since there are no updated drivers that uses Vulkan API: https://www.techpowerup.com/gpu-specs/radeon-hd-4870.c219
You didn't mention what you are going to use this Workstation for but I would suggest, at least, installing a AMD/Windows 10/11 supported GPU card including AMD Professional GPU cards.