Catalyzes the adenosylmethionine-dependent methylation of the exocyclic amino group (N(2)) of guanosine at position 10 of various tRNAs. Acts via a two-step process that leads to the formation of either N(2)-monomethyl (m(2)G) or N(2)-dimethylguanosine (m(2)(2)G) (By similarity). guanosine(10) in tRNA + 2 S-adenosyl-L-methionine = 2 H(+) + N(2)-dimethylguanosine(10) in tRNA + 2 S-adenosyl-L-homocysteine Monomer. Belongs to the methyltransferase superfamily. Trm-G10 family.