You would handle this the exact same way a Vue app would handle it.
All that SSR does is pre-renders the pages HTML content, which is normally done for SEO reasons. Once the browser receives this pre-rendered content, Vue/Nuxt goes through the hydration process to bootstrap the app.
From then on, the application behaves exactly as a normal Vue or Nuxt application would. You can create a button with an @click handler, call a method, and have that method post to your server.
The only tricky thing you need to be aware of is that when rendering server side, the process does not have access to the window object, so attempting to directly reference it, or using plugins/libraries that reference it, is a big nono. For these, you'd need to use <no-ssr></no-ssr>, or check that window exists.