ECDSA Verification - Last Steps Reaching the end of an ECDSA signature verification, we need to compute multiple point multiplication and addition as following: $R=(x_R, y_R) = u_1G + u_2Q$, where $u_1$ and $u_2$ are scalars, $G$ is the curve generator and $Q$ is the public key point. As we're using elliptic curve operations we're computing our curve operations $\mod q$, with $q$ being the field order.

If $R$ is a valid point on the curve and $R \neq\infty$, we need to compute $x_R\mod n$, with $n$ representing the order of the generator, and compare that to the $r$-part of the signature. Why do we need another reduction $\mod q$ if we have already reduced $\mod n$ after the multiplication part?
There are two different primes being used here.

Using your notation, $q$ should be the prime defining the field $\mathbb{F}_q$ over which the curve is defined.

$n$, instead, indicates the (prime) order of your generator $G$.

The coordinates of points, and therefore the computations of scalar multiplications and point addition, are done in $\mathbb{F}_q$.

So, your point $R$ will have coordinates $\bmod q$. But you still need to apply the modular reduction with respect to $n$ to $x_R$ in order to make verification work. 
This is because the $r$-part of the signature is also computed $\bmod n$ and $n$ is allowed to be smaller than $q$.