Change only ONE character in Latex template

Just change the \mathcode of v:

\documentclass{book}

\usepackage{amsmath}
\usepackage{txfonts}

\AtBeginDocument{\mathcode`v=\varv}

\begin{document}

A `v' in math: $v$

A `nu' in math: $\nu$

\end{document}

Output with the \AtBeginDocument line commented out

Output with the \AtBeginDocument line uncommented

Comments

I'd suggest using

\usepackage{newtxtext}
\usepackage[varvw]{newtxmath}

instead of \usepackage{txfonts}. The fonts are essentially the same, but the defects of txfonts are straightened off.

\documentclass{book}

\usepackage{amsmath}
\usepackage{newtxtext}
\usepackage[varvw]{newtxmath}

\begin{document}

A `v' in math: $v$

A `nu' in math: $\nu$

\end{document}

Supposing you are using:

\usepackage{newtxtext,newtxmath}

The following in your preamble replaces your 'v's in mathmode with the \varv symbol in the output.

\DeclareMathSymbol{v}{\mathord}{lettersA}{51}

\documentclass{article}
\usepackage{newtxtext,newtxmath}
\usepackage[T1]{fontenc}
\DeclareMathSymbol{v}{\mathord}{lettersA}{51}
\begin{document}

This is \verb+\varv+ $\varv$

This is v $v$.

This is greek \verb+\nu+ $\nu$.

\end{document}