Quantum Computing (QC) threatens the cryptographic foundations of Cloud Computing (CC), exposing distributed infrastructures to novel attack vectors. This survey provides comprehensive analysis of quantum-safe cloud security, examining vulnerabilities, transition strategies, and layer-specific countermeasures across nine architectural layers (application, data, runtime, middleware, OS, virtualization, server, storage, networking). We employ STRIDE-based risk assessment aligned with NIST SP 800-30 to evaluate quantum threats through three transition phases: pre-transition (classical cryptography vulnerabilities), hybrid (migration risks), and post-transition (PQC implementation weaknesses including side-channel attacks). Our security framework integrates hybrid cryptographic strategies (algorithmic combiners, dual/composite certificates, protocol-level migration), cryptographic agility, and risk-prioritized mitigation tailored to cloud environments. We benchmark NIST-standardized PQC algorithms for performance and deployment suitability, assess side-channel and implementation vulnerabilities, and analyze quantum-safe strategies from leading CSPs (AWS, Azure, GCP). The survey delivers layer-specific threat taxonomies, likelihood-impact risk matrices, and CSP-informed deployment roadmaps for cloud architects, policymakers, and researchers. We identify six critical research directions: standardization and interoperability, hardware acceleration and performance optimization, AI-enhanced security and threat mitigation, integration with emerging cloud technologies, systemic preparedness and workforce development, and migration frameworks with crypto-agility.