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In this work, we perform on-chip quantum random number generation (QRNG) that uses a novel differential amplifier configuration for conjugate homodyne detection. Leveraging separate integrated photonics and integrated analog circuit platforms, we present an alternative method for QRNG. This approach exploits the observable \(\hat{\text{Z}}\), derived from the sum of squared conjugate quadrature distributions which we compare to the traditional single quadrature approach. Utilizing this method, we report a shot noise clearance (SNC) of 25.6 dB and a common mode rejection ratio (CMRR) of 69 dB for our homodyne detection system. We used a variety of design tools to model and predict performance and compare results with our measurements. The realization of our QRNG system consists of a 90{\deg} optical hybrid, a dual differential transimpedance amplifier (TIA), and a field-programmable gate array (FPGA) used for the real-time post-processing to produce a uniform random bitstream. The randomness extraction is implemented using a Toeplitz hashing algorithm and is validated by the National Institute of Standards and Technology (NIST) randomness test suites.