The inclusion of papers in a systematic review should not depend on their statistical significance (it may be at high risk of biasing the review). One of the strengths of SRMA is to be able to pool together results independently of their significance findings if they cope with the inclusion criteria.
Both significant and insignificant correlations should be included in the meta-analysis to avoid selection bias. Furthermore, studies that have reported the association you are studying yet can't be meta-analyzed due to missing data should be synthesized next to your meta-analysis.
Estimation of correlation coefficients (https://handbook-5-1.cochrane.org/chapter_16/16_4_6_3_imputing_correlation_coefficients.htm) as well as using standardized beta coefficients in your correlation meta-analysis (Article On the Use of Beta Coefficients in Meta-Analysis
) may also help increase the power of your meta-analysis.