We propose a new approach to generate quantum-correlated pulse pairs using fission of chirped Manakov solitons. Quantum correlations are studied using linearized quantum fluctuation theory. The fission of chirped solitons produces both intra-pulse and inter-pulse high-level correlations. We found that the collision of vector solitons induces nonzero correlations in orthogonal polarization components of the pulse field. The fission of solitons and subsequent collision of the resulting pulses allow a pulse train with complex correlation pattern to be produced. The generation of pulses with correlated quantum fluctuations offers the possibility of engineering new entangled sources for continuous-variable quantum communication and computation.