Another way to get a Q-filter that is not an ultrafilter is to destroy a Q-point using an $\omega^\omega$-bounding forcing, e.g., Grigorieff forcing with a selective ultrafilter. 

Grigorieff forcing destroys the selective ultrafilter, but because Grigorieff forcing with a P-filter is $\omega^\omega$-bounding, the (formerly selective) filter will remain a Q-filter.