If your computer will boot, then your BIOS (or UEFI on more modern machines) is OK and should not be flashed.
Strictly speaking BIOS/UEFI is firmware not hardware, and is a set of instructions to set the basic parameters of your computer at boot time prior to loading your OS. Once your OS has loaded, your BIOS/UEFI is to all intents and purposes redundant until you shutdown (since all its functions are now supplanted by your OS).
So, if you're able to boot, then you should leave it alone.
The vast majority of BSOD events are caused by flaky drivers, and since the drivers you will be using in Ubuntu will for the most part not be the same as you would be using if you were booting Windows, then there's a very strong possibility that any BSOD problems that you might have had with Windows will not be present in Ubuntu.
Have you tried running Ubuntu from a USB drive yet (before you attempt to install it on your computer) to see whether it is compatible with your hardware. If not, then I strongly recommend that you do so, as this will make no material changes to your machine, and will enable you to see whether you still have the same problems as you had before (or not as the case might be).
Edited by Gary R, 19 August 2016 - 12:20 AM.