How to set some lines right aligned in the align enviroments?

I would place the last three equations in an extra aligned environment as follows:

\documentclass{article}

\usepackage{mathtools}

\DeclareMathOperator{\Ric}{Ric}
\newcommand{\Rico}{\Ric_o}
\DeclareMathOperator{\bi}{bi}

\begin{document}
\begin{align*}
  Q(R) ={}&\bi(R,R)\\
  ={}&\bi(R_I+R_{\Rico}+R_W,R_I+R_{\Rico}+R_W)\\
  ={}&\begin{aligned}[t]
    \bi(R_I,R_I)+2\bi(R_I,R_{\Rico})+2\bi(R_I,R_W)&\\
    +\bi(R_{\Rico},R_{\Rico})+2\bi(R_{\Rico},R_w)&\\
    +\bi(R_W,R_W)&.
  \end{aligned}
\end{align*}
\end{document}

Noticed the changed placement of the alignment charecter in relation to the equals signs and the consequent use of extra braces as ={}& to ensure correct spacing. I have also defined your missing commands with \DeclareMathOperator. I chose to load mathtools instead of amsmath, but that is just personal preference; it loads amsmath anyway.

Here's a couple of options- you didn't tell us what the \Ric command is- please always post a complete MWE :)

I would also use

\newcommand{\bi}{\operatorname{bi}}

for your \bi command, assuming I have interpreted it correctly.

\documentclass{article}
\usepackage{amsmath}
\newcommand{\Rico}{Ric_o}
%\newcommand{\bi}{\mathrm{bi}\,}
\newcommand{\bi}{\operatorname{bi}}

\begin{document}

\begin{align*}
Q(R)&=\bi(R,R)\\
    &=\bi(R_I+R_{\Rico}+R_W,R_I+R_{\Rico}+R_W)\\
    &=\bi(R_I,R_I)+2\bi(R_I,R_{\Rico})+2\bi(R_I,R_W)\\
    &\phantom{{}=}+\bi(R_{\Rico},R_{\Rico})+2\bi(R_{\Rico},R_w)\\
    &\phantom{{}=}+\bi(R_W,R_W).
\end{align*}

\begin{align*}
Q(R)&=\bi(R,R)\\
    &=\bi(R_I+R_{\Rico}+R_W,R_I+R_{\Rico}+R_W)\\
    &=\bi(R_I,R_I)+2\bi(R_I,R_{\Rico})+2\bi(R_I,R_W)\\
    &&&+\bi(R_{\Rico},R_{\Rico})+2\bi(R_{\Rico},R_w)\\
    &&&+\bi(R_W,R_W).
\end{align*}

\end{document}