How to assess the effect of individual difference measure on quadratic fit of a within-subject factor?
By default your 8 levels are probably coded linearly: e.g., 

1, 2, 3, 4, 5, 6, 7, 8


Or

-3.5,-2.5, ..., 2.5,  3.5


To examine quadratic effects you could simply square these values. Interpretation of linear and quadratic parameters will be clearer if you first center the linear variable before squaring.

So you could enter (-3.5)^2 = 12.25; (-2.5)^2 = 6.25, etc.

Equally, some data analysis software will allow you to specify polynomial contrasts (i.e., linear, quadratic, cubic). And in such software you can specify interactions.

In R, you would specify that the 8-level variable is an ordered factor.

In SPSS, the GLM - Repeated measures dialog allows you specify interactions and polynomial contrasts.