Not at all. I lived in Georgia from November 2011 to May 2012 and there were lots of people visiting from nearby Muslim and Arab countries.
In fact Georgia seemed one of the easiest countries to travel to for various Muslim and Arab countries going by conversations I had with people I met while working there in a backpacker hostel. I think in terms of affordability, short travel distance, and openness of both the government and the people.
Just off the top of my head, I met people with either nationality or ethnicity of the following countries which are mostly Muslim or at least I who I met was muslim from there: Azeri, Chechnyan, Indonesian, Iranian, Jordanian, Kuwaiti, Lebanese, Malaysian, Palestinian, Sri Lankan, Syrian, Tamil, and Turkish. Some of those enjoyed it so much they are now regular visitors, others are now living and working there.
Georgia is strongly Orthodox Christian but to me it seemed more out of national pride after getting rid of communism than out of religiosity. But one major region of Georgia, the Adjaran Autonomous Republic, is majority Muslim.
One interesting fact about the capital, Tbilisi, is that it’s one of very few cities on earth to have an operating mosque, synagogue, and Christian church on the same road near each other. As tourists we were welcomed into all of them.
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