If the target sequence exists in the GenBank or Pubmed database, so no need to cite the source because the reviewers will check the sequence matching and the expected size of the PCR product. But, if the target sequence is new or predicted and not submitted to the Pubmed or GenBank so you have to cite the source otherwise it will not be trusted.
It really depends on the paper, but nowadays, most paper actually don't give any Primer sequences anymore. All you will find in the methods section is "primer sequences available upon request"......