According to the answer https://stats.stackexchange.com/a/497785/310702, $\alpha$-quantile sample estimator can be considered as M-estimator with $\rho(y_i,\theta)=\alpha(y_i-\theta)_+ + (1-\alpha)(\theta-y_i)_+$.
From the definition of M-estimator I understand that the M-estimator is of the form $$ \theta^* = \mathrm{argmin}_\theta \sum_{i=1}^n \rho(y_i, \theta) $$ But it's not clear to me why $\rho(y_i, \theta)$ given above is a quantile estimator. Specifically, I'm confuesed by + sign after parentheses.
As @Henry notes, the $+$ sign means the positive part, $x_+ = \max(0, x)$. Hence the loss $\rho$ has derivative:
$$ \frac{\partial\rho}{\partial\theta} = \alpha 1(y > \theta) - (1 - \alpha) 1(y \le \theta),$$
and the expected loss $\mathbb E\rho$ has derivative:
$$ \frac{\partial\mathbb E\rho}{\partial\theta} = \alpha \mathbb P(y > \theta) - (1 - \alpha) \mathbb P(y \le \theta),$$
which is zeroed when $\theta$ is the $\alpha$-quantile of the $Y_i$.