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Quantum algorithms run on quantum computers are qualitatively different from those that run on classical computers. Quantum computing algorithms can be used for several problems in graph theory. Most of the classical algorithms involve searching over some space for finding the shortest-paths problem between two points in a graph and a minimal weight spanning tree. We modified classical Dijkstra's algorithm and implement quantum search instead of classical search, of which it will lead to more efficient algorithm. Also we proposed the structure for non-classical algorithms and design the various phases of the probabilistic quantum-classical algorithm for classical and quantum parts. Finally, we represent the result of implementing and simulating Dijkstra's algorithm as the probabilistic quantum-classical algorithm.