Change equation font to Times New Roman?

Addressing your questions in turn:

1. To change only the math-mode font, while leaving the text-mode font unchanged (cmr, right?), you could load either the newtxmath package or the mtpro2 (short for MathTime Professional II) package. (Note that the latter package isn't free; however, a subset of the fonts, called the "lite" set, provides basic lettering capabilities -- but not "fancy" math fonts such as fraktur, caligraphic, curly, and blackboard bold styles -- and is free.) The mtpro2 font features "optically sized" glyphs for first-level sub- and superscript materials (aka "scriptstyle"-level materials).

   • Separately, if you happen to use LuaTeX or XeTeX, you may want to give the "XITS Math" font package a try.
   • If you happen to be a font purist: be aware that these font packages technically give you Times Roman, not Times New Roman. And if you're not a font purist? You may be excused for not noticing any differences -- they are are quite subtle.

2. In math-mode, be careful about making some symbols both bold and italics. In mathematics, each font shape and weight is generally understood to impart some special meaning. For sure, do not combine bolding and italics just for emphasis. Both newtxmath and mtpro2 provide methods for changing the slant of both lowercase and uppercase greek letters.

3. To make all displayed equations be set flush-left rather than centered, use the document class option fleqn.

4. How to set the contents of equations more compactly? Aaah, some would opine that MS Word's math mode is far too compact, leading to frequent overlaps of letters on the baseline and those in superscript or subscript position. Good news for you: Both newtxmath and mtpro2 render math expressions more compactly than does TeX's default math font (Computer Modern).

Putting these things together and simplifying some of your math code along the way leads to the following modified form of your MWE. The main substantive changes I've made are to replace the equation environment with an align environment and to replace the inner split environment with a bmatrix environment.

\documentclass[fleqn]{article}
\usepackage{amsmath}
\newcommand\QSJ{\mathit{QSJ}} % define a couple of acronyms
\newcommand\CML{\mathit{CML}}
\usepackage{newtxtext,newtxmath}
\begin{document}
\begin{equation}
\begin{split}
L_{\CML}( \theta ) 
&= \Biggl(\, \prod_{g  = 1}^{\QSJ - 1} 
             \prod_{g' = g + 1}^{\QSJ} 
      P\left( [ y ]_g = [ m ]_g,\ [ y ]_{g'} = [ m ]_{g'} \right)  
   \Biggr)\\
&= \Biggl(\, \prod_{g  = 1}^{\QSJ - 1} 
             \prod_{g' = g + 1}^{\QSJ} 
   \begin{bmatrix}
   \phantom{-}\Phi_2 ({\tilde \vartheta }_g,{\tilde \vartheta }_{g'},\nu_{gg'}) 
          - \Phi_2 ({\tilde \vartheta }_g,{\tilde \phi }_{g'},\nu _{gg'})\\
          - \Phi_2 ({\tilde \phi }_g,{\tilde \vartheta }_{g'},\nu _{gg'}) 
          + \Phi_2 ({\tilde \phi }_g,{\tilde \phi }_{g'},\nu _{gg'})
   \end{bmatrix} \Biggr)
\end{split}
\end{equation}
\end{document}

---

For comparison, here's how the equations look like if the times (for text font) and mtpro2 (for math font) packages instead of the newtxtext and newtxmath packages. The overall look is very similar, but the output produced with help of the newtxmath package is a bit more compact.

You can use a package that loads times for math mode, for example mathptmx, txfonts or the new newtxfont (for the last one, for the text use \usepacakge{newtxtext} and for the math \usepackage{newtxmath}.

For bold math use \usepackage{bm} and then \bm{} for the part you want in bold.

To left align equations use the fleqn option of the amsmath package, i.e., \usepackage[fleqn]{amsmath}.