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; }
}