Lucius Malfoy was one of the governors of Hogwarts during the events of the Chamber of Secrets
Mr. Lucius Malfoy, a governor of Hogwarts School of Witchcraft and Wizardry, where the enchanted car crashed earlier this year, called today for Mr. Weasley’s resignation. “Weasley has brought the Ministry into disrepute,” Mr. Malfoy told our reporter. “He is clearly unfit to draw up our laws and his ridiculous Muggle Protection Act should be scrapped immediately.”
Harry Potter and the Chamber of Secrets, Chapter 12, The Polyjuice Potion
But he got fired.
Hogwarts was back to normal with only a few, small differences — Defense Against the Dark Arts classes were canceled (“but we’ve had plenty of practice at that anyway,” Ron told a disgruntled Hermione) and Lucius Malfoy had been sacked as a school governor.
Harry Potter and the Chamber of Secrets, Chapter 18, Dobby’s Reward
Based on the books, Lucius Malfoy does not get the post back next year.
“Keep talking, Malfoy, and I’ll give you a real injury,” snarled Ron.
“– he’s complained to the school governors. And to the Ministry of Magic. Father’s got a lot of influence, you know. And a lasting injury like this” — he gave a huge, fake sigh — “who knows if my arm’ll ever be the same again?”
Harry Potter and the Prisoner of Azkaban, Chapter 7, The Boggart in the Wardrobe
Couple of years later, he was in Azkaban on valid charges.
“The Dementors have left Azkaban,” said Malfoy quietly. “Dad and the others’ll be out in no time...”
Harry Potter and the Order of the Phoenix, Chapter 38, The Second War Begins
So it is unlikely that he ever got the post back. But there is no canonical evidence to support this claim.