Do I need a copyright permission if the title of the song and its singer was mentioned to the song I have written?

There are several other questions you should be asking first. For example, is the track title or singers name a registered trademark? If so you would likely need permission, unless your use falls into an exception. Also are the track or singer's name being used in a way that would be derogatory or dilute their value? If so it's likely a violation. In the end you should contact a qualified entertainment attorney who can obtain all the relevant information before giving you an answer.

Mere reference without any actual use of the work is generally allowed as fair use of the trademarks mentioned, rather than the actual copyrighted work itself.

Since titles and names can't be trademarked or copyrighted, you don't have to worry about violating those laws. However, each artist has the right to control the use of his or her name. You would be better off referring indirectly to both the song and the artist's initials.

You should discuss the specifics with a lawyer who knows music law and intellectual property rights law. The things you might be violating are right of publicity or trademark. These are complex issues and to answer your question, the exact facts are needed.