Praise be to Allaah.
There is nothing wrong with using both dates, so long as the Hijri date is written first, followed by the equivalent Gregorian date. The Hijri date is based on the lunar months, which are clear and can be readily observed by anyone who has eyes to see, but the Gregorian months do not have any clearly visible signs and can only be known through calculations. Hence the Islamic sharee’ah uses the Arabic months for the dates of fasting, Hajj, I’tikaaf and so on. Beginning with the Hijri date is also a manifestation of the symbols and features of Islam among those who do not know them. And Allaah knows best.