Is there a specific supplication that is recommended to be said on the first night of Rajab, such as "O Allah, bless us in Rajab and Sha'ban and enable us to reach Ramadan"?

There is no authentic hadith about the virtues of the month of Rajab. It is not distinguished from other months except that it is one of the Sacred Months. The well-known hadith, "O Allah, bless us in Rajab and Sha'ban and enable us to reach Ramadan," has a very weak chain of narration and is not authentically attributed to the Prophet, peace be upon him. As for a Muslim asking his Lord to enable him to reach Ramadan and supplicating for that, there is no harm in it.