You can use \mathversion{bold} to switch to the bold math font, and \mathversion{normal} to again switch to normal math. You've to use fonts which come with a bold math font, such as New Computer Modern, Concrete Math, XCharter Math, Erewhon Math, Kp Math, XITS Math, etc. Usually these fonts, except for NewCM, don't have full coverage of the glyphs as compared to the normal weight math font.
\documentclass[a4paper]{article}
%\documentclass[preview, margin=5pt, varwidth, convert={density=1000, convertexe={magick}}]{standalone}
\usepackage{microtype}
\usepackage{unicode-math}
\setmainfont[
SizeFeatures={
{Size=-8, Font=NewCM08-Book.otf,
ItalicFont=NewCM08-BookItalic.otf,
BoldFont=NewCM10-Bold.otf,
BoldItalicFont=NewCM10-BoldItalic.otf,
SlantedFont=NewCM08-Book.otf,
BoldSlantedFont=NewCM10-Bold.otf,
SmallCapsFeatures={Numbers=OldStyle}},
{Size=8, Font=NewCM08-Book.otf,
ItalicFont=NewCM08-BookItalic.otf,
BoldFont=NewCM10-Bold.otf,
BoldItalicFont=NewCM10-BoldItalic.otf,
SlantedFont=NewCM08-Book.otf,
BoldSlantedFont=NewCM10-Bold.otf,
SmallCapsFeatures={Numbers=OldStyle}},
{Size= 9-, Font = NewCM10-Book.otf,
ItalicFont=NewCM10-BookItalic.otf,
BoldFont=NewCM10-Bold.otf,
BoldItalicFont=NewCM10-BoldItalic.otf,
SlantedFont=NewCM10-Book.otf,
BoldSlantedFont=NewCM10-Bold.otf,
SmallCapsFeatures={Numbers=OldStyle}}
},
SmallCapsFeatures={Numbers=OldStyle},
BoldSlantedFont=NewCM10-Bold.otf,
SlantedFont=NewCM10-Book.otf,
BoldItalicFont=NewCM10-BoldItalic.otf,
BoldFont=NewCM10-Bold.otf,
ItalicFont=NewCM10-BookItalic.otf,
SlantedFeatures={FakeSlant=0.25},
BoldSlantedFeatures={FakeSlant=0.25}
]{NewCM10-Book.otf}
\setmathfont[
StylisticSet={3},
BoldFont = NewCMMath-Bold.otf
]{NewCMMath-Book.otf}
\AddToHook{bfseries}{\boldmath} % Correct macro by egreg
\begin{document}
\noindent This is some normal text. Now for some normal display math.
\[
\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \exp(2 \symup{\pi i} \xi x) \symup{d}x.
\]
\textbf{Now some bold text and bold display math.}
\mathversion{bold}
\[
\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \exp(-2 \symup{\pi i} \xi x) \symup{d}x.
\]
\mathversion{normal}%
Now some again normal display math.
\[
\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \exp(-2 \symup{\pi i} \xi x) \symup{d}x.
\]
Math gets bolded automatically in a chapter or section header, or when you use math in a \verb|\textbf{}| environment.
\textbf{Some bold text $\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \exp(-2 \symup{\pi i} \xi x) \symup{d}x$.}
\section*{Header $\int_{-\infty}^{+\infty} f$}
\end{document}

\mathversion{bold}.\makeatletter \g@addto@macro\bfseries{\boldmath} \makeatotherwhich automatically makes math in titles bold.\g@addto@macromethod, you can do\AddToHook{bfseries}{\boldmath}.\bfseriesis a robust command, so you're redefining the wrong one.