Comparison of two correlations

I want to correlate stock price of one company and stock indice, taking the price at the end of the day for a period of six years. After that, I want to compare the link for two time series, non successif, using the two same variables. More precisely, I want to know if the stock price is significantly more related of the stock indice in period 1 than in period 2.

For now, I did a Pearson correlation for the two times series and to compare the two I used the Fisher r-to-z transformation.
I'm not sure if it's the good method due to fact that I'm not certain if I can considerate my two correlations as being independant of each other.

Thank you in advance.