Is it permissible to fast the first day of Dhu al-Qi'dah with the intention of completing the six days of Shawwal fasting, after the mistaken belief that it was still from Shawwal?

If you intended to fast certain days without making a vow, and you missed some of them, then there is nothing upon you. However, if you made a vow to fast certain days in Shawwal and you missed some of them until Shawwal ended, then make up for the day you missed.