It's possible to tweak the code. But this produces ugly tables… IMHO it's far better to use the booktabs style (as pointed out in the link you provided; for instance, you can check this answer).
\documentclass{article}
\usepackage{ifthen} % to set up complex condition
\usepackage{float} % to modify position specifier
\usepackage[
pipeTables,
tableCaptions,
]{markdown}
\makeatletter
\renewcommand*{\fps@table}{H} % to set up tables' position specifier to "H"
% Add vertical delimiter
\def\markdownLaTeXReadAlignments#1{%
\addto@hook\markdownLaTeXTableAlignment{|}%
\advance\markdownLaTeXColumnCounter by 1\relax
\if#1d%
\addto@hook\markdownLaTeXTableAlignment{l}%
\else
\addto@hook\markdownLaTeXTableAlignment{#1}%
\fi
\ifnum\markdownLaTeXColumnCounter<\markdownLaTeXColumnTotal\relax\else
\addto@hook\markdownLaTeXTableAlignment{|}%
\expandafter\@gobble
\fi\markdownLaTeXReadAlignments}
% Add horizontal delimiter
\def\markdownLaTeXRenderTableRow#1{%
\markdownLaTeXColumnCounter=0%
\ifnum\markdownLaTeXRowCounter=0\relax
\markdownLaTeXReadAlignments#1%
\markdownLaTeXTable=\expandafter\expandafter\expandafter{%
\expandafter\the\expandafter\markdownLaTeXTable\expandafter{%
\the\markdownLaTeXTableAlignment}}%
\addto@hook\markdownLaTeXTable{\markdownLaTeXTopRule}%
\else
\markdownLaTeXRenderTableCell#1%
\fi
% modifications here
\ifthenelse{\(\markdownLaTeXRowCounter>0\relax \AND \markdownLaTeXRowCounter<\markdownLaTeXRowTotal\relax\)}{%
\addto@hook\markdownLaTeXTable\markdownLaTeXMidRule
}{}
%%
\advance\markdownLaTeXRowCounter by 1\relax
\ifnum\markdownLaTeXRowCounter>\markdownLaTeXRowTotal\relax
\markdownInfo{\the\markdownLaTeXTable}
\markdownInfo{\the\markdownLaTeXTableEnd}
\the\markdownLaTeXTable
\the\markdownLaTeXTableEnd
\expandafter\@gobble
\fi\markdownLaTeXRenderTableRow}
\makeatother
\begin{filecontents*}{./example.md}
This is a table:
| Right |
Left |
Default |
Center |
| 12 |
12 |
12 |
12 |
| 123 |
123 |
123 |
123 |
| 1 |
1 |
1 |
1 |
: Demonstration of pipe table syntax.
\end{filecontents*}
\begin{document}
\markdownInput{./example.md}
\end{document}

As you can see, lines are not correctly connected.
Edit
You can format the header by includind the following code before the \makeatother (modify the header style to fit your needs):
\def\headerStyle#1{{\sffamily\bfseries #1}}
\def\markdownLaTeXRenderTableCell#1{%
\advance\markdownLaTeXColumnCounter by 1\relax
\ifnum\markdownLaTeXColumnCounter<\markdownLaTeXColumnTotal\relax
\ifnum\markdownLaTeXRowCounter=1\relax
\addto@hook\markdownLaTeXTable{\headerStyle{#1}&}%
\else
\addto@hook\markdownLaTeXTable{#1&}%
\fi
\else
\ifnum\markdownLaTeXRowCounter=1\relax
\addto@hook\markdownLaTeXTable{\headerStyle{#1}\\}%
\else
\addto@hook\markdownLaTeXTable{#1\\}%
\fi
\expandafter\@gobble
\fi\markdownLaTeXRenderTableCell}
booktabs, how would one work with that &pipeTables? Also, is there a way of formatting the header? – Nathaniel Bubis Jun 03 '21 at 16:25booktabsdocumentation – NBur Jun 03 '21 at 16:38