Вы можете просто использовать имя слота.
this.$slots.slotName // named slot
this.$slots.default // default slot
Пример из api :
<blog-post>
<h1 slot="header">
About Me
</h1>
<p>Here's some page content, which will be included in vm.$slots.default, because it's not inside a named slot.</p>
<p slot="footer">
Copyright 2016 Evan You
</p>
<p>If I have some content down here, it will also be included in vm.$slots.default.</p>.
</blog-post>
Vue.component('blog-post', {
render: function (createElement) {
var header = this.$slots.header
var body = this.$slots.default
var footer = this.$slots.footer
return createElement('div', [
createElement('header', header),
createElement('main', body),
createElement('footer', footer)
])
}
})