This minimal example, will produce the following PDF:
\documentclass[12pt]{abntex2}
\usepackage[utf8]{inputenc}
\usepackage{newfloat}
\DeclareFloatingEnvironment[fileext=loq,placement={!hbtp},
name=Frame,within=chapter,listname=List of Frames]{board}
% \addto\captionsbrazil{\renewcommand{\boardname}{Quadro}}
% \addto\captionsbrazil{\renewcommand{\listofboardsname}{Lista de Quadros}}
\newif\ifenglish\englishfalse
% \englishtrue
\newcommand{\lang}[2]{\ifenglish#1\else#2\fi}
\ifenglish
\PassOptionsToPackage{language=english}{biblatex}
\PassOptionsToPackage{brazil,main=english,spanish,french}{babel}
\else
\PassOptionsToPackage{language=brazil}{biblatex}
\PassOptionsToPackage{main=brazil,english,spanish,french}{babel}
\fi
\begin{document}
\section{\lang{English Section}{Seção em Português}}
\listofboards*
\lang{Section contents.}{Conteúdo da seção.}
\begin{board}[htb]
\centering
\caption{\lang{Subtitle}{Legenda}}
\begin{tabular}{|c|}
\hline
\lang{Content}{Conteúdo}\\
\hline
\end{tabular}
\fonte{\lang{Author}{Autor}}
\end{board}
\end{document}
Where the captions are mixed with English. If you uncomment the lines \addto\captionsbrazil, Latex will throw the error:
LaTeX Error: \listofboardsname undefined.
I had used something like \addto\captionsbrazil on this other issue How to override command on my preamble, after its redefinition by my class with renewcommand?, and it fixed the problem. How can I use \addto\captionsbrazil with the newfloat package board environment I just created?

\listofboardsnamebut\listboardname(since the existing one for the "List of Figures" is called\listfigurenameand not\listoffiguresname). So the code in your question works fine with this replacement. – Apr 03 '19 at 19:53newfloatdocumentation, I just added a ticket for that: https://gitlab.com/axelsommerfeldt/caption/issues/51 – Apr 03 '19 at 19:57