According to [this .pdf file](http://www.math.uni-bonn.de/people/koepke/PreprintsOrdinals_computations_and_models_of_set_theory.pdf) the definition is this:

Consider the canonical ordering on $\mathsf{Ord\times Ord}$:
$$(\alpha,\beta)\prec(\gamma,\delta)\iff\begin{cases}
\max\lbrace\alpha,\beta\rbrace\lt\max\lbrace\gamma,\delta\rbrace & \lor \\\
\max\lbrace\alpha,\beta\rbrace=\max\lbrace\gamma,\delta\rbrace\land\alpha\lt\gamma&\lor\\\
\max\lbrace\alpha,\beta\rbrace=\max\lbrace\gamma,\delta\rbrace\land\alpha=\gamma\land\beta\lt\gamma
\end{cases}$$

The pairing function, if so, $G(\alpha,\beta)=\operatorname{otp}\lbrace(\gamma,\delta)\in\mathsf{Ord\times Ord}\mid(\gamma,\delta)\prec(\alpha,\beta)\rbrace$.