If we want to use this test to legitimately remove (potential) outliers from a dataset, we should keep in mind that

our data has to be normal distributed,

and that we are not supposed to use this test more than once the same data set.

In my opinion, the Dixon Q-test should only be used with great caution, since this simple statistic is based on the assumption that the data is normal distributed, which can be quite challenging to predict for small sample sizes (if no prior/additional information is provided).

Personally, I would use the Dixon Q-test to only detect outliers and not to remove those, which can help with the identification of uncertainties in the data set or problems in experimental procedures. Intuitively, this is quite similar to an approach of identifying samples that have a large standard deviation.

For example, if I tested ~1000 chemical compounds in some sort of activity assay - each compound 5 times, I would mark compounds that contain Q-test outliers for re-testing, because there might have been some problem in the measurement procedure that could have caused this inconsistency.