Disaggregation of Open Radio Access Network (O-RAN) offers greater transparency, flexibility, and cost-effectiveness; however, there is a risk of an increase in the exposure of the connecting interfaces – O1, O2, A1, F1, E2, etc. The evolving quantum computing makes the current ciphers employed in these interfaces vulnerable — emphasizing the importance of developing a quantum-resistant mechanism to ensure long-term secure communication. The strategy is to use Post Quantum Cryptography (PQC), which can make communication secure during the quantum era. This work investigates the incorporation of PQC in O-RAN — specifically targeting the Midhaul (F1) interface between Central Unit (CU) and Distributed Unit (DU) — Disaggregated and Control User Plane Separation (CUPS) architectures. NIST-standardized post-quantum algorithms — Module-Lattice-Based-Key-Encapsulation-Mechanism (ML-KEM), Module-Lattice-Based Digital Signature Algorithm (ML-DSA), along with Classical algorithms AES-256-GCM through Hash-Based Message Authentication Code (HMAC) based Key Derivation Function (HKDF) — provide a strong and robust security to the proposed work, making the communication quantum resistant by safeguarding from classical and quantum attacks. The analysis is performed multi-fold, demonstrating the feasibility of incorporating PQC with the Disaggregated and CUPS O-RAN — considering Latency, Capacity, Reliability, and Resource Efficiency — providing a foundational work for the quantum-safe O-RAN infrastructure that ensures long-term security.