Moments

Around which point have you taken the moments?

Spring P

That's a good start.
Given that the sign is in equilibrium, and therefore, around P, clockwise moments = anti-clockwise moments, could you find the value of the force on spring Q (i.e. the force which provides the moment through the point where Q is attached to the sign)?

I did (0.65x18) - (0.15x18)-Q = 0
Q=9
e=F/k = 9/240 = 0.0375

Where are you getting (0.15x18) from?
Remember, you're only taking moments from around (i.e. relative to) the point where spring P attaches to the sign.

Oh, I thought that they were the anticlockwise moments that spring Q would provide the sign with. What other moments would be around spring P?

(0.65x18) - (0.15x18) - Q = 0

It might be worthwhile checking your understanding of moments again.

If (0.15x18) is the moment caused by spring Q, about P, then what is Q where you've written

?

Remember that moment = force x perpendicular distance from point.
0.15m is the perpendicular distance between point Q and the sign's centre of mass. 18N is the weight of the sign.
So (0.15m x 18N) actually gives the moment, about Q, caused by the weight of the sign. But as we're taking moments about P, not Q, you don't need to care about this moment at all.