I’m putting this one up quickly so the info is out there. If you come across this and need more info, just comment on the post.

Set up the editor:

      // Create editor
      this.editor = new Quill('#' + this.id, {
        modules: {
          toolbar: {
            container: [
              [{'customStyles': this.styleList}],
              [{'align': []}],
              ['bold', 'italic', 'underline', 'strike'],
              [{'list': 'ordered'}, {'list': 'bullet'}],
              [{'color': []}, {'background': []}],
              ['blockquote', 'code-block'],
              [{'indent': '-1'}, {'indent': '+1'}],
              ['clean']
            ],
            handlers: {
              'customStyles': function (index) {
                vm.applyCustomStyle(index)
              }
            }
          }
        },
        theme: 'snow'
      })

      // Add custom style dropdown
      const dropdownPickerItems = Array.prototype.slice.call(document.querySelectorAll('.ql-customStyles .ql-picker-item'))
      dropdownPickerItems.forEach(function (item) {
        item.textContent = vm.customStyles[item.dataset.value]['name']
      })
      document.querySelector('.ql-customStyles .ql-picker-label').innerHTML = 'Styles' + document.querySelector('.ql-customStyles .ql-picker-label').innerHTML

Inject the styles:

      injectStyles () {
        let Block = Quill.import('blots/block')
        let css = ''

        // Register all custom styles
        for (let i = 0; i < this.customStyles.length; i++) {
          let className = this.customStyles[i]['class']
          let element = this.customStyles[i]['element']
          let style = this.customStyles[i]['style']

          // Set up Quill Blot
          class Blot extends Block {
            static create () {
              let node = super.create()
              if (className) {
                node.setAttribute('class', className)
              }
              return node
            }
          }

          Blot.blotName = this.stylePrefix + i
          Blot.tagName = element
          Quill.register(Blot)

          // Create style list for dropdown
          this.styleList.push(i)

          // Create CSS
          let name = (className) ? element + '.' + className : element
          css += '#' + this.id + ' ' + name + '{' + style + '}\n'
        } // Next i

        // Set master CSS
        this.customCSS = css
      },
      applyCustomStyle (index) {
        if (!index) { index = '0' }
        this.editor.format(this.stylePrefix + index, true)
      }