Are you sure the update exe is correct?
Did you run a md5sum on it and do you have the md5sum to compare with?
Also wine is not windows, so it could be that the update exe is incompatible with wine.
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My first reaction is always to check the download. If it isn't available you have to trust the download.
Also the website only specifies the window versions supported.
Wine isn't mentioned there.
So I am afraid you will have to use windows to do this operation.
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Biosupgrade functionality I have seen are always exe files. The manufacturer of you PC has developed it so.
If the manufacturer of your PC would developed a package to do it in Linux, you could run it, but no manufacturer has shown interest in doing this.
The upgrade site has also a possibility for technical assistance, I have seen. Maybe you could seek assistance there, but I am afraid the answer will be 'use windows'
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There will be a stand-alone DOS executable that will run from a DOS prompt.
The problem with BIOS updates is that they need low level acces to the BIOS. I use Windows as well as Linux and I would ALWAYS boot into DOS to do BIOS updates.
Hiren's boot CD is always worth getting as it has so many useful tools on it including DOS, a Mini Windows or two, Linux and a very useful install of PloP that will let you boot from just about anything.
"Just think of it as leaving early to avoid the rush" - T Pratchett