When we state the model, the error is usually the thing which measures the accuracy of the model. I.e if we model variable $y$ with model $M$, the error is $y-M$. Hence when you state the model there is no error there. Suppose your model $M$ is $f(X)$. Then
$$y = f(X)+\varepsilon$$
For general $f$ this type of model statement generalizes very nicely to the model
$$y = E(y|X) + \varepsilon,$$
since informally $E(y|X)$ is basically $f(X)$ for some $f$. Furthermore error $\varepsilon$ has the following nice property:
Now quantile regression specifies the model for $\tau$-th quantile:
The conditional quantile function $Q_y(\tau|X)$ is defined as
Again, since we condition for $X$, in general case we get that there must exist some $g_\tau$ which satisfies the equation. Note that in this case we model a different function of $y$, conditional quantile, not the conditional expectation.
This does not preclude us from defining
$$\varepsilon = y - Q_y(\tau|X), $$
$$y = Q_y(\tau|X) + \varepsilon, $$
but the error term now does not have nice properties. The conditional quantile function for $\varepsilon$ would be:
which ensures only that the $\tau$-th quantile of $\varepsilon$ when $\tau$-th quantile is used to model $y$ is zero.
In both cases we can have a specification error, i.e. that for example linear hypotheses
$$f(X) = X\beta,$$
$$g_\tau(X) = X\beta(\tau)$$
do not capture the true $f$ or $g_\tau$. But then we explicitly state that one model is the true one and another is approximatio and the true model would not have the error in its definition.