Will a CT of chest....Show calcium build up in an artery? I heard of a full body CT showing that in someone, their heart. Would a CT of chest show it?

Yes. Ct of the chest is not tailored to quantitate calcium score of the coronary arteries, but the amount of calcium seen can suggest the degree of disease. This finding usually leads to cta or heart cath if severe.
Not really. You would need a cardiac ct scan which can show calcium build up in the coronary arteries, but it does expose you to about a years worth of radiation in one test, so choose wisely. And, in the absence of symptoms, you might not be able to get your insurance to cover it.