You shouldn't upgrade a BIOS without a good reason. If the upgrade process fails, you may have to buy a new motherboard.
You need to research your particular motherboard at the ASUS web site to see what BIOSes are available and you need to find out what BIOS you have right now. You should be able to see your BIOS version shown on screen briefly during the boot process. If not, you should be able to find it with a program such as Speccy.
You may find some mention online at ASUS.com or elsewhere that your board is known to have that double-boot problem and that a particular BIOS is supposed to fix it.
I probably would not upgrade the BIOS just on the hope that it might help. It depends on how badly it bothers you.
I tried to find your board at ASUS and failed. It may be that it is so old that it is no longer supported. If that is the case, you probably won't be able to change the BIOS.