I can't test because I don't own mtpro2.sty but perhaps this will help
\documentclass[numbers]{book}
\usepackage{amsmath}
\usepackage[T1]{fontenc}
\usepackage{times}
\usepackage{mtpro2}
\makeatletter
\@tfor\x:=\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu
\do{\count@\numexpr\x-"7000\relax
\expandafter\mathchardef\x=\count@}
\makeatother
\begin{document}
without \verb+\mathrm -->+
$\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu$ \bigskip
with \verb+ \mathrm -->+
$\mathrm{\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu}$\bigskip
with \verb+ \mathbf -->+
$\mathbf{\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu}$
\end{document}
UPDATE. Here is version with intelligent Greek letters. I can not test the mtpro2 with bold math. I could test the lite version on overleaf.
For the full version uncomment the commented out parts in \expandafter\edef\x...
.
\documentclass{book}
\usepackage{amsmath}
\usepackage[T1]{fontenc}
\usepackage{times}
\usepackage[lite]{mtpro2}
\AtBeginDocument{\setbox0\hbox{$\mathrm{\xdef\MATHRMFAM{\the\fam}}%
\mathbf{\xdef\MATHBFFAM{\the\fam}}$}}
\makeatletter
\@tfor\x:=\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu
\do{\edef\y{\expandafter\expandafter\expandafter
\@gobble\expandafter\string\x}%
\expandafter\let\csname ORIG\y\expandafter\endcsname\x
\expandafter\edef\x{\noexpand\ifnum\noexpand\fam=\noexpand\MATHRMFAM\space
\expandafter\noexpand\csname up\y\endcsname
%\noexpand\else
%\noexpand\ifnum\noexpand\fam=\noexpand\MATHBFFAM\space
%\expandafter\noexpand\csname bmup\y\endcsname
\noexpand\else
\expandafter\noexpand\csname ORIG\y\endcsname
\noexpand\fi
%\noexpand\fi
}%
}%
\makeatother
\begin{document}
without \verb+\mathrm -->+
$\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu$ \bigskip
with \verb+ \mathrm -->+
$\mathrm{\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu}$\bigskip
with \verb+ \mathbf -->+
$\mathbf{\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu}$\bigskip
with \verb+ \mathit -->+
$\mathit{\alpha\beta\gamma\delta\epsilon\varepsilon\zeta\eta\theta\vartheta\iota\kappa\lambda\mu\nu}
\end{document}
Produces on overleaf (I didn't even install lite version of mtpro on my disk)

The \mathit
test is for comparison. With full mtpro and the commented-out lines uncommented you should see (if I believe info on macro names from doc) the upright bold Greek letters when using \mathbf
.
\boldsymbol
, unfortunately author has used so many places\mathrm
and\mathbf
, i cant able to trace wherever is occuring, so i need a error to throw those greek characters inside the\mathrm/\mathbf