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Does anyone know what the default method is for adjusting for multiple comparisons is in the TukeyHSD function? (e.g. BH, Bonferroni, etc.)

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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.

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