Start enumerating an item in the same line after a exercise number

You can create a new list, say eq, with the resume option for doing this.

\documentclass[a4paper,brazil, 12pt]{report}%{\article}
\usepackage{amsmath,amsfonts,amscd,bezier}
\usepackage{amssymb}
\usepackage{babel}
\usepackage[latin1]{inputenc}
\usepackage[dvips]{graphicx}
\usepackage{color}
\usepackage{hyperref}
\usepackage[shortlabels]{enumitem}
\usepackage{multicol}
\usepackage{tasks}

\newlist{eq}{enumerate}{1}
\setlist[eq]{resume,label={\arabic*.}}%

\begin{document}

\begin{eq}\item 
    \begin{enumerate}[$(a)$]
    \item Prove that...
    \item Next ...
    \item 
\end{enumerate}
\end{eq}

\begin{eq}\item 
    \begin{enumerate}[$(a)$]
    \item Prove that...
    \item Next ...
    \item 
\end{enumerate}
\end{eq}

\end{document}

Hereis a solution which plays with leftmargin= and before= keys:

\documentclass[a4paper,brazil, 12pt]{report}%
\usepackage{amsmath, amsfonts, amscd, bezier}%
\usepackage{amssymb}
\usepackage{babel}
\usepackage[utf8]{inputenc}
\usepackage{graphicx}
\usepackage{color}
\usepackage[shortlabels]{enumitem}
\usepackage{multicol}
\usepackage{tasks}
\usepackage{hyperref}
\newcounter{theeq} \setcounter{theeq}{0}
\newcommand{\eq}[1]{\noindent%
    \refstepcounter{theeq}\textbf{\arabic{theeq}} {\hspace{-0.1cm{{.}}}} }
\newcommand{\eqz}{{\setcounter{eq}{1}} {{\arabic{eq}}} \refstepcounter{eq}\hspace{-0.1cm{{.}}}}

\begin{document}

\eq. \begin{enumerate}[$(a)$, leftmargin=3.2em, before=\vspace*{-\glueexpr\topsep + \baselineskip}]
    \item Prove that...
    \item Next ...
\end{enumerate}

\end{document} 

Edit:

Another possibility would be to define your own enumerate-like list, like this:

\newlist{enumexp}{enumerate}{1}
\setlist[enumexp,1]{$(a)$, leftmargin=3.2em, before=\vspace*{-\glueexpr\topsep + \baselineskip}}

and code in the document body:

\eq. =\begin{enumexp}
\item ...
.........
\end{enumpexp}