As for month of birth, yes, all persons in samples after 1981 have their birth month recorded in the variable BIRTHMO. Additionally, birth year (BIRTHYR) was recorded for all persons starting with the 1997 sample.
As for month of study, I assume you are referring to the month of the NHIS interview? This information is recorded in the variable INTERVWMO, starting with the 1997 sample. A relatively small number of persons were interviewed in the year after the sample year, as captured by the variable INTERVWYR.