Is it common for people who are infected with hiv to develop bell's palsy within the first few weeks after being exposed to the hiv virus?

Any thing can happen: If there's any chance of exposure to HIV, the individual must seek counseling and testing though appropriate channel. See the PCP or contact the local health department for free testing, This is a very serious concern that required immediate professional attention. There's also available post-exposure treatment that may apply depending on timing.