I'm not quite sure what you mean by an "open-ended questionnaire." I think it might be an instrument presenting questions to which the respondent is supposed to produce self-generated statements. If that's the case, then I don't see why you couldn't use thematic analysis. Of course, some of the prompts may elicit similar themes from most participants, but that's okay - you're looking for patterns & also individual differences, I assume.

There is nothing particularly quantitative about an open-ended question that was part of a larger questionnaire. Of course it will produce shorter, more specific answers than full-scale qualitative interview, but it is still qualitative data.

Whether that data is "rich" enough to produce themes is another matter. If the answers are indeed short and relatively concrete, then you should look into content analysis, especially with regard to coding manifest (rather than latent) content.

to go further, the best could be to present an example of your open-ended answer. But at a first glance, these answers are qualitative and not quntitative data. 

I would agree with the above observations, and if you are looking for simple forms of qualitative techniques, you might want to try the attached.

Chapter Mapping Research Methods

my opinion is that analysis of the  answers of  open ended questions couldn't be qualitative method, because in qualitative methods our intention to achieve "deep" understanding of interviewee, the goal that doesn't achieve in the  open ended questions, you could do content analysis and categorize your quantitative data  

My experience of using open ended questions in a questionnaire is that they have to be administered carefully, otherwise you may get little in the way of responses, and little to analyse. In my experience questionnaires tend to be better with quantitative data, and interviews/focus groups work better with collecting in depth qualitative data. Descriptive stats could be used to analyse the quanti' data and there are numerous methods for quali' data analysis. I tend to use grounded theory, open and axial coding could be useful starting point depending on how far you intend to take the analysis. See my paper on obesity on research gate for a detailed explanation of how I applied GT analysis for a published article.

Thank you all for your useful answers