No: Shi et al., analyse over 14,000 genes and find no evidence to suggest that lots of genes specific to human brains have evolved more rapidly than those in Chimpanzee/Macaque brains.
There are a few genes which do seem to have evolved more rapidly in human brains (e.g. FOX P2 – implicated in language production, and ASPM ([1], [2]), which causes microcephaly when it goes wrong.
However, Shi et al’s study suggests that despite these (notable) exceptions, the genes involved in human brain evolution have undergone stronger purifying selection than those in the chimpanzee lineage.