This paper investigates the dynamical connection function in $ f(Q) $ cosmology beyond the coincident gauge using Hubble data and Gaussian Processes. The authors explore three branches of connections that satisfy torsionless and curvatureless conditions, parameterized by a new dynamical function $ \gamma $. They express the redshift dependence of $ \gamma $ in terms of the $ H(z) $ function and the $ f(Q) $ form and parameters, and reconstruct it using 55 $ H(z) $ observation data. In the case where ordinary conservation law holds, they reconstruct the $ f(Q) $ function, which is well described by a quadratic correction on top of the Symmetric Teleparallel Equivalent of General Relativity (STEGR). They also consider two well-studied $ f(Q) $ models: the square-root and exponential ones. In both cases, they reconstruct $ \gamma(z) $ and show that its inclusion is favored compared to the $ \Lambda $ CDM paradigm and the same $ f(Q) $ models under the coincident gauge, indicating that $ f(Q) $ cosmology should be studied beyond the coincident gauge. The study uses Gaussian Processes to reconstruct $ H(z) $ from observational data, and then uses this to reconstruct $ \gamma(z) $. The results show that $ \gamma $ deviates from zero, indicating a deviation from the coincident gauge. The inclusion of $ \gamma $ improves the quality of the fits to observations compared to both the $ \Lambda $ CDM paradigm and the $ f(Q) $ models under the coincident gauge. The results suggest that $ f(Q) $ cosmology should be studied beyond the coincident gauge to better understand the dynamics of the universe.This paper investigates the dynamical connection function in $ f(Q) $ cosmology beyond the coincident gauge using Hubble data and Gaussian Processes. The authors explore three branches of connections that satisfy torsionless and curvatureless conditions, parameterized by a new dynamical function $ \gamma $. They express the redshift dependence of $ \gamma $ in terms of the $ H(z) $ function and the $ f(Q) $ form and parameters, and reconstruct it using 55 $ H(z) $ observation data. In the case where ordinary conservation law holds, they reconstruct the $ f(Q) $ function, which is well described by a quadratic correction on top of the Symmetric Teleparallel Equivalent of General Relativity (STEGR). They also consider two well-studied $ f(Q) $ models: the square-root and exponential ones. In both cases, they reconstruct $ \gamma(z) $ and show that its inclusion is favored compared to the $ \Lambda $ CDM paradigm and the same $ f(Q) $ models under the coincident gauge, indicating that $ f(Q) $ cosmology should be studied beyond the coincident gauge. The study uses Gaussian Processes to reconstruct $ H(z) $ from observational data, and then uses this to reconstruct $ \gamma(z) $. The results show that $ \gamma $ deviates from zero, indicating a deviation from the coincident gauge. The inclusion of $ \gamma $ improves the quality of the fits to observations compared to both the $ \Lambda $ CDM paradigm and the $ f(Q) $ models under the coincident gauge. The results suggest that $ f(Q) $ cosmology should be studied beyond the coincident gauge to better understand the dynamics of the universe.