Before showing menu or splash-screen, Grub2 load something like a console for some messages like errors, It shows for about 200ms but the cursor is very visible. It is an aesthetic problem but I would appreciate to control this behavior and hide the cursor.
This appears to be extremely hardcoded. I initially seeked for a solution for the same problem as I faced it in my UEFI-enabled laptop to which I installed a BIOS version of Ubuntu. I looked everywhere and I hoped someone would answer your question, but no one did and I doubt anyone will (with a solution), so I started to look for alternatives to Grub.
I converted my hard drive from MBR to GPT, converted my installation to UEFI, and then started booting UEFI-mode to Ubuntu. Since an UEFI Ubuntu still uses Grub, believe it or not, that goddamn cursor was still there, only now superimposed on the Lenovo logo screen!! Amazing.
So in the end I ditched Grub entirely by booting via EFI stub kernel loader. Now there is no cursor anymore. I recommend you also look for alternatives to Grub if you're determined to get rid of the cursor. Sorry this is a non-answer to the question, but I ravaged the web for an answer and nobody seems to even recognize it as a thing, so... I don't think there's an option to switch it off or something (I even tried theming Grub to make it black font on black background: as I didn't even multiboot this wouldn't be a problem. Still didn't work).
Short answer: Without compiling your own grub2 boot loader - you can't.
Longer answer: Compiling your own grub is not that hard and it is doable pretty fast. However you need to add your own interrupt handler INT 10h routine to hide blinking cursor (function code AH=01h) for which you need to know basics of assembly code. It has been tested and it works. You can more easily get silent boot with some other boot loader such as syslinux. However I couldn't manage to get UEFI secure boot with syslinux therefore I stayed with grub2 boot loader.