grammaticality - Does the sentence "How do you like Painting Exhibition?" make any sense?

I pressume it was the first part of the question (How do you like) what puzzled you, as it is a strange construction to English learners.

That construction might be a subject of some debate since the word "How" asks about the quality (good, bad, acceptable...) of something or the way (nicely, acceptably...) something is done. And yet, the real question here is about a quantity (i.e., "How much do you like it?"). That may be confusing for non-native speakers, but this construction is pretty common in everyday speech, and perfectly grammatical. Actually, it is also pretty common to hear "I like X better than Y", meaning you like it more. So once again, quality and quantity seem to mix together, for some reason unkown to me.

The only aspect that is not grammatical about your sentence is the absence of a determiner. The P in Painting shouldn't be capitalized, and it should be preceded by a determiner, preferably a "the".

So, to answer your question, it surely makes sense and is grammatically correct, if stated as follows:

How do you like the painting exhibition?