Is it permissible to mention a country other than the real country on a YouTube page to give the channel a sense of closeness and credibility?

Lying is forbidden except in exceptional cases. As for double entendre (tawriyah) and insinuation (ta‘rid), they are permissible when there is a need, such as concealing the country of residence of the website owner if there is a benefit in doing so, provided that they are not used to unlawfully acquire someone else's property or infringe upon their rights, and that the website's content does not include anything that violates Islamic law.