Four members of the public and two police officers were taken to Glasgow Royal Infirmary for treatment after the incident.
Police said a woman has been charged with alleged road traffic offences.
The vehicle which was involved in the Cowcaddens Road incident was a black Audi TT.
Police Scotland confirmed it had been called to a "disturbance".
As Freshers' Week started, the area was busy with new students and promoters from pubs and clubs.
A Police Scotland spokeswoman said: "Around 12.30pm on Monday, a car being driven on Cowcaddens Road struck four pedestrians and a stationary police car with two police officers in it.
"The six people have been taken to Glasgow Royal Infirmary where they are being treated for minor injuries."