Short answer : Maybe.
Longer answer : It depends.
If you have the welding current passing -through- the alternator, then yes you will fry things (circuitry, bearings, etc). So don't 'ground' through the alternator itself.
If you for some reason loop the welding cables around or 'across' the alternator, you may induce voltages in the windings of the alternator or the internal circuitry (if the alternator is so equipped). Which may fry things.
If you are running AC or a TIG with high-frequency start, you may induce voltage in the alternator windings or internal circuitry. So as with any use of AC or high-frequency, actually "ground" the alternator to an actual electrical "ground" (not the welding circuit 'ground' aka "work lead").
If you get some slag or weld spatter into the windings (or somehow the bearing, etc), then you can fry or damage things.
Probable damage? Probably not.
Absolutely not damage anything? I couldn't tell.
So the longer answer is maybe and "It depends".
Really cautious approach : Tack the bracket together on the bench or in place without the alternator. Attach alternator and check fit and alignment. Adjust as needed. Then do final welding without alternator and after the bracket cools do final 'inspection' and check fit and alignment.
