Let $M_2$ be the spaces of conjugacy classes of quadratic maps in End($\mathbb P^1$). By Milnor, $M_2$ is biholomorphic to $\mathbb C^2$. And he introduces the complex lines $\mathrm{Per}_1(\lambda)$ in $M_2$, which is the set of conjugacy classes with $\lambda$ as a multiplier of some fixed point. We show that $\mathrm{Per}_1(\lambda)$ has infinitely many postcritically finite conjugacy classes if and only if $\lambda=0$. There are two main ingredients: (1) we show that the bifurcation measures for the two marked critical points are different (2) we apply the equidistribution theorem (Baker-Rumely, Favre-Rivera-Letelier, Chambert-Loir, 2006) to show that: if there are infinitely many postcritically finite conjugacy classes, they must equidistribute with respect to the above two bifurcation measures.