Monster truck should not be hyphenated there at all. But to answer your question, hyphenated compound modifiers are hyphenated whether they’re in quotation marks or not.
Here "High Mountain" is the actual name of a mountain in Alaska. Do you hyphen this?
Or do you hyphen --High-Mountain--like you would I am scared of the high-mountain people (people living in high mountains) and say --He likes the High-Mountain villages (c