The IgM antibodies could usually reach a peak within the few weeks after the infection period of Borrelia bacteria that could collapse after a time period f 4-6 months after the significant infection.
You could also estimate that the IgG antibodies, present due to Lyme’s disease slowly develop and begins to show in about 4-6 weeks after any exposure and that may peak in the duration of 4-6 months after the exposure.