Eh, no.

Goethe was born into a Lutheran family and was well versed in biblical matters. He even attempted his own translation of the bible which is reflected in a scene in Faust. This, on the other hand, doesn't imply that he viewed himself as a Christian. He wasn't known to be a regular churchgoer and his writings reveal, generally speaking, pantheistic views. For him religion wasn't a matter of institutions but of "sentiment" (I don't know an appropriate word that encapsulates the 18th/19th century meaning of "Gefühl"). One probably has to delve deeper in how the Enlightenment viewed religious concepts to get a better understanding of that point.

This non-dogmatism made it easy to be generally open to other religions' ideas and Goethe synthesized views across a wide range of sources. Which brings me to the likely source of the misconception that Goethe was Muslim which is his book "West-östlicher Divan". It is a collection of poems he wrote after reading Persian poetry by Hafez who himself was unorthodox at his time. The lyrical subject of these poems identifies as Muslim and the whole work is one that emphasizes the affinities of Orient and Occident but is also critical of certain aspects of Islam, for example treatment of women, alcohol prohibition and the very concept that the Koran is considered to be not divinely inspired but is itself divine:

To summarize: He was deeply affected by the poetry in the Islamic word and the poetic nature of the Koran itself but he wasn't a Muslim.