Let $\mu_{{M},{\mathcal{D}}}$ be a self-affine measure generated by an expanding real matrix $M=\left(\begin{array}{cc}a&e\\f&b\end{array} \right)$ and the digit set $\mathcal{D}= \{(0,0)^t, (1,0)^t, (0, 1)^t, (1, 1)^t\}$. In this paper, we consider that when does $L^2(\mu_{M,\mathcal{D}})$ admit an infinite orthogonal set of exponential functions? Moreover, we obtain that if $e=f=0$ and $a, b\in\{\frac{p}{q},p,q\in 2\mathbb{Z}+1\}$, then there exist at most 4 mutually orthogonal exponential functions in $L^2(\mu_{M,\mathcal{D}})$, and the number 4 is the best possible.