Thai newspapers reported that the slogans "destroying the country" and "die for the country" had been sprayed on the car.
Ekapol Tawichwongchaikul, police major general, said: "The car was badly damaged but the tank was unscathed."
Police are questioning the 60-year-old driver, who was taken to hospital with broken ribs.
"He was probably drunk or half asleep. We found no weapon or bomb in his car," Ekapol said.
Activists have organised small protests against the coup, but there has been no significant opposition to the coup leaders who say they have the support of the population.
Political gatherings of more than five people are banned under the current martial law, but it is loosely enforced.
The military said on Friday it was still investigating Wednesday's arson attacks on five schools in central Thailand, a stronghold of the former prime minister.
Thaksin is in exile in London with members of his family, while investigations continue into corruption allegations against him and his cabinet colleagues.
The coup leaders, grouped in the Council for Democratic Reform, promise to hand over power to a civilian government before their self-imposed two-week deadline expires on October 4.
On Friday, CDR leader General Sonthi Boonyaratglin said he expected an interim constitution to come into force on Sunday, after which the name of a new prime minister would be submitted for royal approval.
Human rights groups are worried that the army's decision to stay on as a Council for National Security (CNS), while a final charter is worked out over the next nine months, will undermine its independence.
The interim charter gives the CNS the power to replace the civilian government, but Sonthi said it would only be used in exceptional circumstances such as if the prime minister died.
"I can assure you it is impossible that we will control the government," he said.
"We will be the government's tool to keep peace."
Sonthi declined to name Thaksin's successor, but Thai media on Saturday continued to focus on respected former army chief Surayud Chulanon, a senior royal adviser.