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Phase transitions in dissipative quantum systems are intriguing because they are induced by the interplay between coherent quantum and incoherent classical fluctuations. Here, we investigate the crossover from a quantum to a classical absorbing phase transition arising in the quantum contact process (QCP). The Lindblad equation contains two parameters, $ω$ and $κ$, which adjust the contributions of the quantum and classical effects, respectively. We find that in one dimension when the QCP starts from a homogeneous state with all active sites, there exists a critical line in the region $0 \le κ< κ_*$ along which the exponent $α$ (which is associated with the density of active sites) decreases continuously from a quantum to the classical directed percolation (DP) value. This behavior suggests that the quantum coherent effect remains to some extent near $κ=0$. However, when the QCP in one dimension starts from a heterogeneous state with all inactive sites except for one active site, all the critical exponents have the classical DP values for $κ\ge 0$. In two dimensions, anomalous crossover behavior does not occur, and classical DP behavior appears in the entire region of $κ\ge 0$ regardless of the initial configuration. Neural network machine learning is used to identify the critical line and determine the correlation length exponent. Numerical simulations using the quantum jump Monte Carlo technique and tensor network method are performed to determine all the other critical exponents of the QCP.