Does the Quarantine Specialist prevent Epidemic in connected cities?
Yes, she cancels the placement of any disease cubes on a city she is in and connected cities.

Note that this does not "prevent the Epidemic"... when an epidemic card is drawn, you still fully deal with it. It's just that during the "infect" step of the epidemic, she can prevent the extra infection.

From the rules:


  The Quarantine Specialist prevents both outbreaks and
  the placement of disease cubes in the city she is in
  and all cities connected to that city. She does not affect
  cubes placed during setup.