We report on a unique photonic quantum source chip highly integrating four-stage photonic elements in a lithium niobate (LN) waveguide circuit platform, where an aperiodically poled LN (APPLN) electro-optic (EO) polarization mode converter (PMC) is sandwiched between two identical type-0 PPLN spontaneous parametric down-converters (SPDCs), followed by an EO phase controller (PC). These core nonlinear optic and EO building blocks on the chip are systematically characterized stage by stage to show its high performance as an integrated quantum source. The APPLN EO PMC, optimally constructed by a genetic algorithm, is characterized to have a broad bandwidth (>13 nm), benefiting an efficient control of broadband type-0 SPDC photon pairs featuring a short correlation time. We demonstrate an efficient conversion of the |VV› photon-pair state generated from the first PPLN SPDC stage to the |HH› state through the APPLN EO PMC stage over its operating bandwidth, a broadband or broadly tunable polarization-entangled state can thus be possibly produced via the superposition of the |VV› state generated from the other PPLN SPDC on the third stage of the chip. Such a state can be further manipulated into two of the Bell states if the relative phases between the two polarization states can be properly modulated through the EO PC on the fourth stage of the chip. Such a multifunction integrated quantum photonic source chip can be of high value to developing a compact, efficient, and high-speed quantum information processor.