240mm is the maximum size of graphics card allowed for the PC-Q11. Pushing and pulling will not help. :(
You are correct, the next mini-ITX case up that would fit it is the PC-Q28, and there is a bit of a size and price difference between the two cases. However, if you can find some of the older models, such as the PC-Q08 and PC-Q18, your card will fit in there and you can probably get it for a lower price.