Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on each modular curve
X
0
(
N
)
X_0(N)
of genus 2, 3, 4, or 5 whose Mordell–Weil group has rank 0. In this paper we do the same for the
X
0
(
N
)
X_0(N)
of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of
N
N
are 37, 43, 53, 61, 57, 65, 67, and 73.
The main tool used is a relative symmetric Chabauty method, in combination with the Mordell–Weil sieve. Often the quadratic points are not finite, as the degree 2 map
X
0
(
N
)
→
X
0
(
N
)
+
X_0(N)\to X_0(N)^+
can be a source of infinitely many such points. In such cases, we describe this map and the rational points on
X
0
(
N
)
+
X_0(N)^+
, and we specify the exceptional quadratic points on
X
0
(
N
)
X_0(N)
not coming from
X
0
(
N
)
+
X_0(N)^+
. In particular, we determine the
j
j
-invariants of the corresponding elliptic curves and whether they are
Q
{\mathbb {Q}}
-curves or have complex multiplication.