Yes, the discussion linked to by @ArminNiaki gives lots of good information. Let me just make a couple of points.

• Correction methods like Bonferroni (FWER and FDR methods) are for p-values. The way your question is written, it is unclear if you are talking about correcting p-values or correlation coefficients (r). If p-values, you will decide whether you want to make a correction. If correlation coefficients, there is no need to do any correction.
• In most cases Bonferroni is excessively conservative, and another p-value correction method will probably be better.
• I would say that for multiple correlations, a p-value correction is usually not done. This is because the correlations are usually presented for informational purposes or for purposes that don't really involve making a decision. Ultimately, whether or not to apply a correction is up to the analyst (and reader, and reviewers).

Qing,

I think the following link will answer your questions sufficiently:

https://www.researchgate.net/post/Is_it_necessary_to_do_correction_to_significance_level_while_conducting_Pearson_correlation_coefficients_between_different_continuous_variables

Hope this helps.

Yes, the discussion linked to by @ArminNiaki gives lots of good information. Let me just make a couple of points.

• Correction methods like Bonferroni (FWER and FDR methods) are for p-values. The way your question is written, it is unclear if you are talking about correcting p-values or correlation coefficients (r). If p-values, you will decide whether you want to make a correction. If correlation coefficients, there is no need to do any correction.
• In most cases Bonferroni is excessively conservative, and another p-value correction method will probably be better.
• I would say that for multiple correlations, a p-value correction is usually not done. This is because the correlations are usually presented for informational purposes or for purposes that don't really involve making a decision. Ultimately, whether or not to apply a correction is up to the analyst (and reader, and reviewers).

Since you ask about Bonferroni I assume you want to look at the p values. p-values for correlation coefficients are of limited use, especially if you have more than a few tens of samples. All the p-value does is tell you if the correlation is significantly different from 0. With 200 or so samples correlations below 0.01 are significant, but a correlation value of this magnitude tells you that each variable it is not contributing much to the variation of the other.