You can get negative Q values as the Q denotes the energy output (enthalpy change) of a particular substance. If you imagine the tables we're provided for Hess' cycles, some of the values are negative for the same reason (representing a negative enthalpy change).
Or at least as far as I'm aware (I've been awarded full marks for negative values and having gone over mark schemes it states that a mark should be deducted for positive values)...
Hope this helps,