Collis Hussey, 2003 claim that qualitative has low reliability and high validity, whereas quantitative data has high reliability and low validity. If it is so, then how it can be explained and proved?
That will depend on the purpose of your research, and the availability or otherwise of the data needed. Data about the future, for instance, is likely to be unavailable; whereas experiential qualitative data might be available.
Kahneman and Flyvbjerg, both of whom are here on Researchgate, have covered the benefits of qualitative research as compared to quantitative.
I did qualitative research in "Human Factors in Ship Design" - and it shows that qualitative research has high reliability and high validity. It is in agreement with the quantitative approach.
One of the reason why qualitative research seems to be "not reliable" is probably that the researcher has not completely finished with her/his data collection. She/he should not stop until she/he reach "reliability".
Another reason is probably "over generalization". One may study one sample, but would like to generalize for four :-)