Need system specs and graphics card model.
Try running HW monitor and monitor the temperatures. Then hook in the second monitor. Keep the system running for a few minutes. Now remove the second monitor and see if the display comes back on (you might have to wait for a few minutes. If and when it comes back on, read the highest temperature of the GPU. If it doesn't turn on, feel the graphics card when both monitors are plugged in and see if it heats to an alarming point.