If a child turns blue without having a temper tantrum, then it definitely requires investigation, especailly of the heart. However, the fact that it is related to episodes of fever and shaking (?cold fever or rigors) does not fit with a heart condition, but more with a possible severe infection. My advice would be to share your concerns with your paediatrician who will need to take a careful history and do at least an ultrasound (echo) test of the heart.
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