How do I change which monitor my computer uses at boot?

Which screen gets BIOS output during the boot process is up to the graphics card. Some graphics cards output to all displays simultaneously, some will only output to whichever port it considers to be "port 1". Most of the time they're not labeled. In the case of multiple graphics cards, the display goes to whichever one the BIOS detects first. Some BIOSes have a setting that lets you determine if an onboard vs. discrete card should be treated as the "primary".

One thing that you should know about TVs specifically is that most of them do not power on their HDMI port unless the TV is actually set to that input. So if your computer is connected to HDMI2, but the TV is set to HDMI1, the computer may not even see that there is a screen attached at all until you switch the TV over to HDMI2.

Also... some BIOSes do not detect whether or not a screen is attached to the primary display and some do. In the cases where it does, "port 1" might change depending on whether the TV is on or set to the right input. In cases where it doesn't, output may go to the HDMI port whether there's anything connected to it or not, and you may not see anything on either screen.

Unfortunately, GRUB loads very early in the boot process before any drivers. It relies on the BIOS to put its menus up on the screen, so you're totally at the mercy of the hardware at that point.

So short answer to your question is "maybe" :-)

I know the original is question is old, but today I met with the same problem. I have a Gigabyte GTX 1060, and I'm using a TV on HDMI, and a Monitor on Display port. The VGA has 3 Display Ports, 1 HDMI and 1 DVI. Fortunately my problem is solved after I changed the used Display Port (from the farthest to the closest from HDMI). No BIOS change is needed.

Now my monitor shows the boot screen properly. If you didn't try it already, I hope this will help :/.