It is recommended to complete the Quran during Ramadan. The following excerpt from Imam al-Nawawi’s Etiquette With the Quran touches on many of the legal issues and etiquette related to completing the Quran.
The Etiquette of Completing the Quran
There are several issues concerning completing the recitation of the Quran [khatma].
WHEN TO COMPLETE: The first etiquette concerns its timing. As previously mentioned, it is preferable for someone reciting the Quran on his own that [he complete it] during Prayer. Another opinion is that it is preferable for it to occur in the two rak‘a of the obligatory Morning Prayer, and in the two sunna rak‘as of the Sunset Prayer, and the two rak‘as of the Morning Prayer is better. In another opinion, it is preferable for the completion [khatma] in the beginning of the day for one turn [of recitation], and at the end of the day for next turn.