Entirely possible, but not with the little baby engraving machine you showed.
YouTube 'Hublot factory' and see what the factories themselves use, to give you an idea.
When you have the machine, you need a gen case and Solidworks ($5K US) to draw a case with measurements you get using a vernier caliper. You'd need to deconstruct the entire watch.
Then you'd need a converter to machine code, then a converter to G-code, trust me, it's a pain in the ass.