  In this work we study the Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies
with arbitrary spatial curvature for the symmetric teleparallel theories of
gravity, giving the first presentation of their coincident gauge form. Our
approach explicitly starts with the cosmological Killing vectors and constructs
the coincident gauge coordinates adapted to these Killing vectors. We then
obtain three distinct spatially flat branches and a single spatially curved
branch. Contrary to some previous claims, we show that all branches of
connection in the covariant approach can be studied in this gauge-fixed
formalism, which offers certain conceptual advantages. Interestingly, we find
that the flat and negative spatially curved solutions in $f(Q)$ gravity can be
seen as equivalent to the metric teleparallel $f(T)$ theories, demonstrating a
deeper connection between these theories. This is accomplished by studying the
connection equation of motion, which can be interpreted as a consistency
condition in the gauge-fixed approach. Finally, we discuss the role of
diffeomorphism invariance and local Lorentz invariance in these geometric
modifications of gravity.
