If OBS is not running on the GPU driving the monitor in question, OBS cannot capture it.
A workaround is to use an HDMI capture device/card. Unfortunately if the external device/monitor is USB-driven, that is not an issue; the GPU is generally integrated with the usb-monitor. About the only possibility would be fitting the external monitor-unit with a capture device (not generally feasible) or using something like a same-system NDI capture and transfer to obs-ndi.