The Prophet (ﷺ) forbade fasting Rajab.
Sunan Ibn Majah Book 7, Hadith 106
← Previous Hadith “A man came to the Prophet (ﷺ) and said: ‘Which fasting is better after the month of Ramadan?’ He said: ‘The month of Allah which is called Muharram.’”

Sunan Ibn Majah Book 7, Hadith 105 Next Hadith → “Fast Shawwal.” So he forsook the sacred months and he continued to fast Shawwal until he died.

Sunan Ibn Majah Book 7, Hadith 107