"lunar month" is correct and usable in written English.
You can use it when referring to the amount of time it takes for the moon to circle the earth, which is usually around 28 days. For example, "It only took 10 lunar months for the spacecraft to reach the moon's orbit."
*** From "Red Sorghum" (1993) The ninth day of the eighth lunar month, 1939.
Derwent Pulse will take place over the lunar month in October.
Chinese New Year, or Spring Festival, is celebrated on the first day of the first lunar month, which usually falls in late January or early February.
It runs for 12 days, counting from the second day of the lunar month of Chaitra, in April or May.
It falls on the last full moon day of the lunar month at the end of winter.
As early as Pope Sixtus I, some Christians had set Easter to a Sunday in the lunar month of Nisan.
Giridhar's Upanayana (sacred thread ceremony) was performed on Nirjala Ekadashi (the Ekadashi falling in the bright half of the lunar month of Jyeshtha) of 24 June 1961.
Tano, Korean holiday celebrated on the fifth day of the fifth lunar month to commemorate the start of summer and to honour spirits and ancestors.
ʿUmar started the first year ah with the first day of the lunar month of Muḥarram, which corresponds to July 16, 622, on the Julian calendar.
In January, for example, Saudi religious authorities abruptly announced, a day earlier than expected, the start of the Muslim lunar month of Dhu'l Hijja.
The female menstrual cycle has been shown to have its mean length indistinguishable from the synodic lunar month [ 29, 30], and women whose cycles average 28 30 days tend to have their menstrual period phase-linked to the lunar month [ 31, 4].
