The following was helpful.

https://stackoverflow.com/questions/59407003/set-focus-text-field-inside-dialog-when-dialog-opened

By accessing $refs after a short delay when opening the dialog, it worked successfully.

watch: {
   dialog: function(value) {
      if (value) {
        setTimeout(() => {
          this.$refs.name.focus();
       }, 200);
     }
   }
}