Quantum computing becomes more of a reality as time passes, bringing several cybersecurity challenges. Modern cryptography is based on the computational complexity of specific mathematical problems, but as new quantum-based computers appear, classical methods might not be enough to secure communications. In this paper, we analyse the state of the Galileo Open Service Navigation Message Authentication (OSNMA) to overcome these new threats. This analysis and its assessment have been performed using OSNMA documentation, reviewing the available Post Quantum Cryptography (PQC) algorithms competing in the National Institute of Standards and Technology (NIST) standardization process, and studying the possibility of its implementation in the Galileo service. The main barrier to adopting the PQC approach is the size of both the signature and the key. The analysis shows that OSNMA is not yet prepared to face the quantum threat, and a significant change would be required. This work concludes by assessing different temporal countermeasures that can be implemented to sustain the system's integrity in the short term.