You have to use the no-math
option to fontspec
:
\documentclass{book}
\usepackage{extarrows}
\usepackage[no-math]{fontspec}
\setmainfont{Times New Roman}
\begin{document}
$Q\xLongrightarrow{\text{Maximum}} P$
\end{document}
But the result is disputable, as Computer Modern math fonts don't combine well with Times.

What happens without the no-math
option is that the upright math family (family 0) is changed to use the normal text font; the equals sign is taken from math family 0, so it doesn't combine with the double arrow taken from Computer Modern Math Symbols.
Should you want to use TeX Gyre Termes Math along with Times New Roman for the text font, some workarounds are necessary, since the developers didn't provide for the double arrows to combine with the equals sign.
\documentclass{book}
\usepackage{extarrows}
\usepackage{unicode-math}
\setmainfont{Times New Roman}
\setmathfont{TeX Gyre Termes Math}
\setmathfont[
range={`=,\Longrightarrow,\Rightarrow,\Longleftarrow,\Leftarrow}
]{Asana Math}
\setmathfont[range=\int]{TeX Gyre Termes Math}
\begin{document}
$Q\xLongrightarrow{\text{Maximum}} P$
$Q\xLongleftarrow{\text{Maximum}} P$
\end{document}

A complete workaround covering also extendable single arrows.
\documentclass{book}
\usepackage{extarrows}
\usepackage{unicode-math}
\setmainfont{Times New Roman}
\setmathfont{TeX Gyre Termes Math}
\setmathfont[
range={
`=,\Longrightarrow,\Rightarrow,\Longleftarrow,\Leftarrow,
\harrowextender,\longrightarrow,\rightarrow,\longleftarrow,\leftarrow
}
]{Asana Math}
\setmathfont[range=\int]{TeX Gyre Termes Math}
\renewcommand{\relbar}{\mathrel{\harrowextender}}
\begin{document}
$Q\xlongrightarrow{\text{Maximum}} P$
$Q\xlongleftarrow{\text{Maximum}} P$
\end{document}
palatino
), and does not loadfontspec
, but concludes to afontspec
bug. – Thérèse Jul 13 '14 at 13:06minimal
class withfontspec
and without it and it should demonstrate my point. – user56799 Jul 13 '14 at 13:13