Is it permissible for someone who paid the zakat on his wealth according to his birthdate, then discovered his error, to change the date to the Hijri calendar while calculating the difference in days, and is there any sin incurred for the error in the first year?

If the zakat reaches its nisab (the minimum amount upon which zakat is due), you must know the date of that according to the Hijri calendar. Zakat must be paid after one full Hijri year has passed on your ownership of the nisab, provided the nisab has not decreased during that year. If you paid zakat late because you calculated it using the Gregorian calendar, then in the following years you must revert to the Hijri date from the time you acquired the nisab. As for the previous delay, we hope that Allah will overlook it due to your ignorance of the ruling. It is not permissible to delay zakat after the completion of the Hijri year, though it is permissible to pay it in advance before the year is complete, according to the majority of scholars.