One can either use a more stringent alpha level, or keep alpha=0.05 but make the p-values bigger...the effect is the same. I'm not sure how to do this in SPSS, but there might be an option to do some type of correction for multiple tests from a correlation matrix.
Here is an example using R. Suppose we have three p-values: 0.001, 0.01, 0.04. A Bonferroni adjusted alpha level would be 0.05/3 = 0.016777, and so the first two p-values would be considered significant, but the third one would not.
If we use the p.adjust function in R we get
and so we would come to the same conclusion (first two p-values are significant (<0.05), but the third one not). The documentation for this function states "The adjustment methods include the Bonferroni correction (‘bonferroni’) in which the p-values are multiplied by the number of comparisons."
I hope this helps.