Blockquotes can be added to using typical Markdown blockquote syntax:

I came across this section of the Hugo help guide when looking at syntax highlighting.

CSS classes can now be added to Markdown blocks such as tables, lists, paragraphs etc.

This only works with Hugo versions 0.81 and up.
To upgrade Hugo: brew upgrade hugo

The example shows the class added using dot notation in curly brackets {.myclass} immediately below the block. You can add styling for the class in the CSS stylesheet. If using Bootstrap then the bootstrap class can be added. I am trying it below using a blockquote as none of my blockquotes in Markdown were rendering as blockquotes.

Configuring markup

https://gohugo.io/getting-started/configuration-markup/

CSS classes can now be added to Markdown blocks such as tables, lists, paragraphs etc.

Trying to style the blockquotes by adding a class using bootstrap blockquote class another line of a blockquote

blockquote attribute here using the blockquote-footer class (bootstrap)

I added the following from the Goldmark configuration on this page and changed block = true from the default block = false.

  [markup.goldmark.parser]
      autoHeadingID = true
      autoHeadingIDType = "github"
      [markup.goldmark.parser.attribute]
        block = true
        title = true

A blockquote in markdown using a shortcode.

> Here is a blockquote using Markdown and a shortcode that uses bootstrap classes.