Is it true that after lord Rama defeated Ravana and declared Vibhishana the king, Vibhishana married Mandodari?
Traditions differ from one kingdom/area to another.
Vanaras follow a tradition of marrying wife of dead brother. That was why Sugriva married Tara, after the death of Vali.
On reaching Kishkinda, in Pushpaka Vimana, Sita says as follows;
सुग्रीवप्रियभार्याभिस्ताराप्रमुखातो नृप || ६-१२३-२४ अन्येषां वानरेन्द्राणां स्त्रीभिः परिवृता ह्यहम् | गन्तुमिच्छे सहायोध्यां राजधानीं त्वया सह || ६-१२३-२५
"O king! I wish to reach the capital City of Ayodhya along with you, accompanied by the wives of other monkey-chiefs with Tara and other beloved wives of Sugreeva."
However, Rakshasas did not follow this tradition.
According to Valmiki Ramayana, Vibhishana becomes king of Lanka, after the death of Ravana. However, it was not mentioned that he married his dead brother's wife, Mandodari.