The data association problem of multi-object tracking (MOT) aims to assign IDentity (ID) labels to detections and infer a complete trajectory for each target. Most existing methods assume that each detection corresponds to a unique target and thus cannot handle situations when multiple targets occur in a single detection due to detection failure in crowded scenes. To relax this strong assumption for practical applications, we formulate the MOT as a Maximizing An Identity-Quantity Posterior (MAIQP) problem on the basis of associating each detection with an identity and a quantity characteristic and then provide solutions to tackle two key problems arising. Firstly, a local target quantification module is introduced to count the number of targets within one detection. Secondly, we propose an identity-quantity harmony mechanism to reconcile the two characteristics. On this basis, we develop a novel Identity-Quantity HArmonic Tracking (IQHAT) framework that allows assigning multiple ID labels to detections containing several targets. Through extensive experimental evaluations on five benchmark datasets, we demonstrate the superiority of the proposed method.