TDerden

showButton () {
    if (!this.article.section || !this.article.title || !this.article.description) {
        return false;
    }

    if (this.$route.name === 'new-article') {
        return true;
    }

    if (this.$route.name === 'update-article' && this.show) {
        return true;
    }

    return false;
}

Решение в лоб:

showButton () {
  if (this.$route.name === 'new-article') {
    return !!(this.article.section && this.article.title && this.article.description);
  }
  if (this.$route.name === 'update-article') {
    return !!(this.show && this.article.section && this.article.title && this.article.description);
  }
}

с выносом в отдельную переменую повторяющихся частей

showButton() {
  var is_sect_title_descr = !!(this.article.section && this.article.title && this.article.description);
  if (this.$route.name === 'new-article')    { return is_sect_title_descr; }
  if (this.$route.name === 'update-article') { return this.show && is_sect_title_descr; }
}