How soon after goat gives birth can you start milking her?

No. Once a cow has given birth and is lactating, you must keep milking that cow--be it once or twice a day--in order for her to keep producing milk. You don't milk cows once after they've given birth and that's it, they get milked for usually 10 months out of the year before being dried up to be ready to calve. Some dairies will inject a form of growth hormone (called Posilac in the US) in their cows to increase production and/or to keep them producing for longer, especially if they're older cows that are unable to breed and the producer doesn't want to cull them out right away.