In order to improve tracking, interference and multipath mitigation performance from that possible with existing signals, a new Global Navigation Satellite System (GNSS) signal is needed that can offer additional degrees of freedom for shaping its pulse waveform and spectrum. In this paper, a new modulation scheme called Quinary Offset Carrier modulation (QOC) is proposed as a new GNSS signal design. The pulse waveforms of QOC modulation are divided into two types: convex and concave waveforms. QOC modulations can be easily constructed by selecting different modulation parameters. The spectra and autocorrelation characteristics of QOC modulations are investigated and discussed. Simulations and analyses show that QOC modulation can achieve similar performance to traditional BOC modulation in terms of code tracking, anti-multipath, and compatibility. QOC modulation can provide a new option for satellite navigation signal design.