May I gently barge into this conversation to offer my view on your question regarding the possibility of doing naturalist/ positivisitc research in HE?
I tend to equate study abroad programmes (to take one example of a topic within internationalisation) with language development, so I was pleasantly surprised to read Earnest, Rosenbusch, Wallace-Williams, and Keim’s (2016) study in which psychology majors from the U.S. spent two weeks in Argentina. They operationalised some constructs including emotional resilience and perceptual acuity into manipulable variables by using Kelley and Meyers’ (1995, cited in Earnest et al, 2015) Cross-Cultural Adaptability Inventory. Means, standard deviations and analysis of variances were derived from the numerical variables. Their results were surprising; “students studying abroad reported decreased flexibility and openness after returning home” (p. 78). An avenue for further investigation, perhaps?
Another quantitative study (Yang, 2015) researched the risk attitude Chinese college students had regarding studying abroad. Yang also used an attitude measuring instrument and performed logistic regression statistics to test the hypotheses.
As for other aspects in internationalisation, devising quantitative studies does not seem so problematic. I hope that this helps.
Earnest, D. R., Rosenbusch, K., Wallace-Williams, D., & Keim, A. C. (2016). Study Abroad in Psychology: Increasing Cultural Competencies Through Experiential Learning. Teaching of Psychology, 43(1), 75–79. doi:10.1177/0098628315620889
Yang, X. (2015). Chinese College Students’ Risk Attitude to Moving Abroad to Study. Social Behavior and Personality, 43(5), 795–802. Retrieved from http://dx.doi.org/10.2224/sbp.2015.43.5.795