SÃO PAULO—Brazilian politicians and rights groups rallied around the country’s first gay activist lawmaker on Friday after he said he was giving up his seat and going into exile following death threats.

Jean Wyllys, a vocal advocate for gay rights who has often clashed with President Jair Bolsonaro, said he feared for his life amid a rise in antigay aggression since the election of Mr. Bolsonaro, an ex-army captain known for his antigay positions.