Does this count as a quantitative research?: word association
You answered your own question. You plan to "quantify the strength of word association." In this instance, yes, your work would be considered quantitative research. 

[Wikipedia] defines quantitative research as "the systematic empirical investigation of observable phenomena via statistical, mathematical or computational techniques" 1. The example you cite does exactly that. 

Yes, everything is quantifiable some way. The greater question is whether or not that quantification has meaning. Yes, you can quantify emotions (countries are compared on happiness scales all the time). How that concept was measured and what is done about it is much more important. 

As evidenced by the comment left "the example you give is peculiar," there is a level of debate that scholars could have about the validity and credibility of your methodology. Whether or not your methods were quantitative is not at question (they were). 

I then add one more question to your question: Does it matter? I would much rather argue the value of your research with you than whether it would be considered qualitative or quantitative in nature.