Does anyone know what the default method is for adjusting for multiple comparisons is in the TukeyHSD function? (e.g. BH, Bonferroni, etc.)
The Turkey HSD (honest significant difference) test is itself a procedure for statistical testing that takes into account multiple comparisons. It's based on the Studentized range distribution of the difference between the maximum and minimum of a set of a number of values (expressed in terms of sample standard deviations). The multiple-comparisons issue is handled by the incorporation of the number of values being compared (mean values, in this case) as a parameter of the Studentized range distribution.
Tukey's HSD attempts to control family-wise error rate so in that sense it is more similar to the Bonferroni correction than to the Benjamin-Hochberg false discovery rate approach. But it is a test of its own and depends on no other correction for multiple comparisons.