Video Card Problems/Z580 Graphics
My question was, is it possible to buy a laptop GPU and install it in a lenovo Z580 when it does not come with one stock?
Generally speaking, "laptop GPUs" are not something that can be purchased as stand-alone upgrades, nor are they upgrades that users can conventionally install. A select few machines may have MXM slots which will accept MXM modules, however these modules are rarely available for commercial purchase (and are often marked up heavily on the used market), and compatibility is not as uniform or universal as we would like.
Ultimately, laptops are designed to be replaced, as opposed to repaired or upgraded, and, this disposable nature inherently limits the options of end-users for hardware upgrades or servicing. It's an unfortunate trade-off (performance/features for mobility). Ultimately my best advice to users is to purchase laptops (or other portable computers) when mobility is required over other functionality (as it will be purchased at the expense of other functionality). If high-performance graphics are a requirement, desktop or workstation hardware is always the best choice (and in many cases the only choice).
Regarding your specific machine:
The Lenovo website lists all available Z580 configurations as based around an integrated Intel graphics controller, and does not list any graphics hardware as an available accessory or upgrade module. You may still try contacting Lenovo regarding such an upgrade (there is a very small chance that your specific machine may support MXM, and that they may perform such upgrades through a factory service center (but will not sell the hardware to end-users)). My honest feeling is that you will not find such an upgrade module for this system, as a quick web search yielded no immediate results (and often upgradeable laptops develop something of a following), but again it may be worth contacting the manufacturer to inquire. Keep in mind that if such an upgrade board were to exist, it would cost substantially more than the performance is worth, and would likely reduce the effective battery life of the system (and run the system hotter).
Lenovo product page for the Z580: