Several computational studies on different water models reported evidence of a phase transition in supercooled conditions between two liquid states of water differing in density: the high-density liquid (HDL) and the low-density liquid (LDL). Yet, conclusive experimental evidence of the existence of a phase transition between the two liquid water phases could not be obtained due to fast crystallization in the region where the phase transition should occur. For the same reason, the investigation of possible transition mechanisms between the two phases is committed to computational investigations. In this work, we simulate an out-of-equilibrium temperature-induced transition from the LDL to the HDL-like state in the TIP4P/2005 water model. To structurally characterize the system relaxation, we use the node total communicability (NTC) we recently proposed as an effective order parameter to discriminate the two liquid phases differing in density. We find that the relaxation process is compatible with a spinodal-like scenario. We observe the formation of HDL-like domains in the LDL phase and we characterize their fluctuating behavior and subsequent coarsening and stabilization. Furthermore, we find that the formation of stable HDL-like domains is favored in the regions where the early formation of small patches of highly connected HDL-like molecules (i.e., with very high NTC values) is observed. Besides characterizing the LDL- to HDL-like relaxation from a structural point of view, these results also show that the NTC order parameter can serve as an early-time predictor of the regions from which the transition process initiates.