The video, a trailer for a new BBC programme which will be aired for the first time on Tuesday, has 2.3 million views on Facebook at the time of writing.
Although this is the first time it has received international attention, the ex-MP's's bridge-building project is not new – she has previously met with both radical Muslims and neo-Nazis.
The issue of prominent women in Danish politics receiving online abuse is also a long-standing one.
Former Socialist People's Party (SF) MP, Cekic, a Kurd who was born in Ankara, moved to Denmark as a child. She qualified as a nurse in 2000 before entering politics in 2005, when she stood for election as an MP. She entered parliament in 2007 but lost her seat in 2015 as SF performed poorly at the polls.
In March this year, she left SF, citing the party's position in support of preventing unaccompanied refugee minors from entering Denmark – a position later reversed by SF.
The 41-year-old Muslim has previously stated her belief that religion and state should be separate. She has also voiced fears that her critical stance against the Turkish government could lead to her being arrested by Ankara.
The BBC sent a camera crew with Cekic when she visited a man called Stefan, who sent her an abusive message on Facebook.
The man is reported by Politiken to have called Cekic “disgusting ISIS vermin and disgusting carrion”.
He also wrote “I hate you and everything your kind stands for”.
The former MP also says in the video that she has been called a “monkey” in hate mails.
At one point, the conversation becomes too much for Cekic, who is seen leaving the room and breaking down into tears.
“Those who were born here we have to take care about [sic.],” says the man, whose first name only is given, confronted by Cekic in the BBC video.
He tells Cekic that he wrote what he did as an “eye-opener”.