As part of my research I have conducted two survey questionnaires where the first one has closed ended questions and the second one has both closed and open ended questions. My doubt was whether to call this a Quantitative or Mixed Methods approach?
A survey questionnaire comprising both open-ended and closed-ended questions—and formed with the intent to integrate the two datasets to develop results and interpretations—can be considered a mixed-method research approach. Closed-ended questions render quantitative data that can be analyzed using statistical methods, while open-ended questions furnish qualitative data that can be analyzed for patterns and themes. By combining both types of questions, researchers could comprehensively understand the problem under investigation. Here is an insightful textbook.
Creswell, J. W., & Plano Clark, V. L. (2018). Designing and conducting mixed methods research (3rd ed.). SAGE. https://us.sagepub.com/en-us/nam/designing-and-conducting-mixed-methods-research/book241842
The answer provided by Mohialdeen Alotumi is good and I want to add just one detail. Consider the difference between "approach" and "research" in your question. You approach is certainly mixed-method because it allows to qualitatively analyze data generated by open-ended question. Whether you actually perform mixed-method research depends on what you actually do with the data.
However I would like to highlight that the first survey which has closed ended questions is only to check if the participant is eligible and has the knowledge to participate in the second survey which has open ended questions.
In this scenario, in my opinion, a sequential explanatory mixed methods research approach would be best suited.
From your reference to 2 surveys, the first closed question survey is part of your sampling methods, checking participant eligibility, and the the second is a qualitative research survey.
This makes your research qualitative as it is seeking knowledge of a person's socially constructed experience.
I had a follow up question to this. If I am using descriptive statistics to analyse quantitative findings and thematic analysis to analyse qualitative findings, then which would be the method for integration?
A survey questionnaire with both open and closed-ended questions can be considered as a mixed methods approach, as it incorporates both quantitative (closed-ended) and qualitative (open-ended) elements into the research design.