This paper presents numerical evidence of a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random guessing for solving 3-SAT and Max-SAT problems. G-QAOA is more efficient and adaptable than Grover's algorithm and outperforms conventional QAOA in sampling all solutions. The study uses classical simulations on thousands of random 3-SAT instances and demonstrates G-QAOA's effectiveness on the IonQ Aria quantum computer for small instances. A single-angle-pair constraint significantly reduces classical computational overhead while preserving the quadratic speedup. The angles in G-QAOA cluster within a small parameter range, making optimization more efficient. G-QAOA requires fewer qubits and circuit depth compared to the standard Grover algorithm, especially at critical densities. The paper also shows that G-QAOA provides fair sampling of solutions, a key advantage for All-SAT and Max-SAT problems. The results indicate that G-QAOA outperforms the X-QAOA in both solution probability and fair sampling, even on noisy devices. The single-angle-pair protocol simplifies parameter optimization, enabling efficient solutions for large instances. The study highlights the potential of G-QAOA for future large-scale applications, reducing classical overhead and addressing challenges like barren plateaus in variational quantum algorithms. The paper concludes that G-QAOA offers a quadratic speedup and fair sampling for 3-SAT and Max-SAT, with promising results for quantum computing applications.This paper presents numerical evidence of a quadratic speedup of the Grover Quantum Approximate Optimization Algorithm (G-QAOA) over random guessing for solving 3-SAT and Max-SAT problems. G-QAOA is more efficient and adaptable than Grover's algorithm and outperforms conventional QAOA in sampling all solutions. The study uses classical simulations on thousands of random 3-SAT instances and demonstrates G-QAOA's effectiveness on the IonQ Aria quantum computer for small instances. A single-angle-pair constraint significantly reduces classical computational overhead while preserving the quadratic speedup. The angles in G-QAOA cluster within a small parameter range, making optimization more efficient. G-QAOA requires fewer qubits and circuit depth compared to the standard Grover algorithm, especially at critical densities. The paper also shows that G-QAOA provides fair sampling of solutions, a key advantage for All-SAT and Max-SAT problems. The results indicate that G-QAOA outperforms the X-QAOA in both solution probability and fair sampling, even on noisy devices. The single-angle-pair protocol simplifies parameter optimization, enabling efficient solutions for large instances. The study highlights the potential of G-QAOA for future large-scale applications, reducing classical overhead and addressing challenges like barren plateaus in variational quantum algorithms. The paper concludes that G-QAOA offers a quadratic speedup and fair sampling for 3-SAT and Max-SAT, with promising results for quantum computing applications.