I would suggest that you update the BIOS since this is usually required as a standard troubleshooting step, but in order to determine whether this is a CPU or motherboard issues you would need to either test another known work processor on that board or test your processor on another board. If this is not possible, you can contact Intel support and request a motherboard replacement, since it is more probable that this is a motherboard issue and this possibility needs to be discarded first.
If you are still facing the problem,
1. Power down the MB completely and then remove the cmos battery. Wait for some time and then reboot->F2-> Load bios defaults. See if this works
- if F2 method is not visible, keeping power button pressed for 3+ sec resets the bios to default.
2. The GFX is built into CPU, however there are other components in the path. To rule out issue with the on-board HDMI connector, which can go bad due to mech stress, you can try to connect the monitor to Dport instead of hdmi port. You will need to borrow/buy an DPORT to HDMI cable.