This study unveils a comprehensive design strategy, intricately addressing the realization of transmon qubits, the design of Josephson parametric amplifiers, and the development of an innovative fully integrated receiver dedicated to sensing ultra-low-level quantum signals. Quantum theory takes center stage, leveraging the Lindblad master and quantum Langevin equations to design the transmon qubit and Josephson parametric amplifier as open quantum systems. The mentioned quantum devices engineering integrated with the design of a fully integrated 45 nm CMOS system-on-chip receiver, weaves together a nuanced tapestry of quantum and classical elements. On one hand, for the transmon qubit and parametric amplifier operating at 10 mK, critical quantum metrics including entanglement, Stoke projector probabilities, and parametric amplifier gain are calculated. On the other hand, the resulting receiver is a symphony of high-performance elements, featuring a wide-band low-noise amplifier with a 0.8 dB noise figure and ~37 dB gains, a sweepable 5.0 GHz sinusoidal wave generator via the voltage-controlled oscillator, and a purpose-designed mixer achieving C-band to zero-IF conversion. Intermediate frequency amplifier, with a flat gain of around 26 dB, and their low-pass filters, generate a pure sinusoidal wave at zero-IF, ready for subsequent processing at room temperature. This design achieves an impressive balance, with low power consumption (~122 mW), a noise figure of ~0.9 dB, high gain (~130 dB), a wide bandwidth of 3.6 GHz, and compact dimensions (0.54*0.4 mm^2). The fully integrated receiver capability to read out at least 90 qubits positions this design for potential applications in quantum computing. Validation through post-simulations at room temperature underscores the promising and innovative nature of this design.