A lookahead branch-and-bound algorithm (LBnB) is proposed for solving the Maximum Quartet Consistency (MQC) Problem where the input is a complete set of quartets on the taxa and the goal is to construct a phylogeny that satisfies the maximum number of given quartets. It integrates a number of previous efforts on exact algorithms, heuristics, and approximation algorithms for the NP-hard MQC problem, and a few improved search techniques, especially a lookahead scheme, to solve the problem optimally. The theoretical running time analysis of the LBnB algorithm is provided, and an extensive simulation study has been well designed to compare the algorithm to previous existing exact algorithms and a best heuristic Hypercleaning. The experimental results on both synthetic and real datasets show that LBnB outperformed other exact algorithms, and it was competitive to Hypercleaning on many datasets.