I have submitted a paper to a journal. Having got no reply (after waiting for more than the period of acceptance) I tried to connect the editor via telephone numbers given on website. It said that the telephone number did not exist. So, I tried to open the website link, it did not open initially and later it connected me to a different site with the same name where it sold hardware parts. Hence I thought it was a fake website and applied to another journal and was accepted. Now it asks me to pay fee to publish paper. Is there any problem if I publish the paper in the second journal?
Based on what you describe, I would see it as exceedingly likely that the journal is indeed fake. I would formally retract the paper at the journal, and afterwards discontinue all dealings with them. They will likely not answer, but I would just assume that the entire submission is dead as soon as you have told them that you wish to retract. The will likely not publish the paper anyway, if you are not paying them a fee (which you will of course not do).
After you have retracted the submission to the first journal, there are no ethical issues with submitting the paper again, this time making sure that the journal is in fact "real". Some advise you to tell the second journal about the "history" of the paper, but I am honestly not convinced that this is ethically or practically necessary. As long as the paper is not under consideration elsewhere (and it is not, as you have retracted the original submission), the new journal does not need to know where you have tried before to get the same material published. For instance, one also does not typically indicate if a paper was rejected before at a different venue.