Software Quality Assurance (SQA) is critical for delivering reliable, secure, and efficient software products. The Software Quality Assurance Process aims to provide assurance that work products and processes comply with predefined provisions and plans. Recent advancements in Large Language Models (LLMs) present new opportunities to enhance existing SQA processes by automating tasks like requirement analysis, code review, test generation, and compliance checks. Simultaneously, established standards such as ISO/IEC 12207, ISO/IEC 25010, ISO/IEC 5055, ISO 9001/ISO/IEC 90003, CMMI, and TMM provide structured frameworks for ensuring robust quality practices. This paper surveys the intersection of LLM-based SQA methods and these recognized standards, highlighting how AI-driven solutions can augment traditional approaches while maintaining compliance and process maturity. We first review the foundational software quality standards and the technical fundamentals of LLMs in software engineering. Next, we explore various LLM-based SQA applications, including requirement validation, defect detection, test generation, and documentation maintenance. We then map these applications to key software quality frameworks, illustrating how LLMs can address specific requirements and metrics within each standard. Empirical case studies and open-source initiatives demonstrate the practical viability of these methods. At the same time, discussions on challenges (e.g., data privacy, model bias, explainability) underscore the need for deliberate governance and auditing. Finally, we propose future directions encompassing adaptive learning, privacy-focused deployments, multimodal analysis, and evolving standards for AI-driven software quality.