Catalysis of the reaction: L-lysyl4- + 3 S-adenosyl-L-methionine = 2 H+ + N6,N6-trimethyl-L-lysyl4- + 3 S-adenosyl-L-homocysteine. This reaction is the successive addition of three methyl groups to the unmethylated lysine residue at position 4 of histone H3, producing histone H3K4me3.
histone H3K4 trimethylase activity