Crowment is a web component, so a Hugo site only needs the embed code in the template that renders a post.
Sign in to Crowment, create a comment widget for your site's domain and copy its app key from Install & Embed.
Find the template that renders a single post in your theme, usually single.html or page.html under themes/<theme>/layouts/. Copy it to the same path under your site's own layouts/ folder. Hugo uses your copy instead of the theme's, and theme updates won't overwrite your change.
{{ .Content }}
<script defer src="https://embeds.crowment.com/comments.js"></script>
<crow-comments app-key="YOUR_APP_KEY"></crow-comments>
Run hugo server and open a post. Each post gets its own thread, identified by its URL path.
partial "comments.html" in the template), you can put the embed code in that partial instead and leave the template alone.page-id: see Which thread a page shows.