How to get Markdown processed content in Jekyll tag plugin

I'm working on a Jekyll tag plugin for my Octopress site to help me make a 'note' element. I just want to be able to highlight a piece of information on my blog as a side note, like this.

The problem is, I can't figure out how to get the contents of this tag to be processed (i.e. Markdown or Textile). The above image is only achieved is I actually make my links with html code. Here is how it ends up turning out when I use markdown in the contents.

In my post, I'm writing the contents of this like so.

{% note %}
This is the third post in my Start to Finish series.  Last time I talked about [Git](/blog/2013/09/25/getting-started-with-git/).
{% endnote %}

Here is my plugin code. It's based off of the image tag code, and there's really not a lot to it.

module Jekyll
  class NoteTag < Liquid::Block
    @title = nil

    def initialize(tag_name, markup, tokens)
      @title = markup
      super
    end

    def render(context)
      output = super(context)
      title = "Note"
      if !@title.empty?
        title += ": #{@title}"
      end
      "</section>\n<div class=\"note\"><span class=\"title\">#{title}</span>#{output}</div>\n<section>"
    end
  end
end

Liquid::Template.register_tag('note', Jekyll::NoteTag)

Do you have any idea how I can use a converter on the contents of this tag? I generally use Markdown for my posts, but I'd like to release this plugin for others so I'd like it to be dynamic just like the rest of Jekyll.

Jekyll 3.x : getConverterImpl is now deprecated

Use find_converter_instance to get the converter :

def render(context)
  text = super
  site = context.registers[:site]
  converter = site.find_converter_instance(::Jekyll::Converters::Markdown)
 _output += "<figcaption>#{converter.convert(_caption)}</figcaption>"