I want to perform significance test for correlation coefficient. However my data is not normally distributed. In my books I have found tests only for the normally distributed data. Is there such test I could apply to data having any distribution?
I am assuming that you are interested strictly in finding a p-value to test $H_0: \rho=0$ and yet want to control Type I error rates. Then you may want to check out this article that discusses 12 tests of correlation, including familiar Pearson and Spearman as well as less known tests based on transformation and resampling approaches. The authors examine which tests are robust to various non-normal distributions across a range of sample sizes. Judging from the shape of your data's distribution and sample size, you can pick the best test.
Anthony J. Bishara, James B. Hittner, Testing the significance of a correlation with nonnormal data: comparison of Pearson, Spearman, transformation, and resampling approaches Psychol Methods, 17(3), 399-417 (2012)
If you're testing that the correlation differs from zero you could do a permutation test (or a randomization test). This is probably the simplest to do.
Alternatively if sample sizes are not small, you could form a bootstrap interval for the correlation and see if it includes whatever value you want to test for a difference from.
If you are prepared to assume a distribution you may be able to derive a test or perhaps simulate the distribution of the correlation under independence.