The adverb 'heavily' is used to express 'how' or 'in what manner'. In the given sentence she sighed when she heard the news which indicates that the news wasn't good. Thus, the adverb 'heavily' is used to express the sad manner in which she sighed.
She sighed when she heard the news. Option A i.e. happy cannot be used here as it is an adjective and we would have to convert it into its adverb 'happily'. It is still incorrect as one does not sigh when one is happy.
When an adjective ending in 'y' is changed into an adverb, the 'y' changes into 'i' and 'ly' is added. Hence, option C is incorrect as the spelling should be 'heavily'.
Option D is also incorrect as a sigh is a long drawn breath and not a quick one also, quick is an adjective which cannot modify the verb sighed.
Therefore, the correct answer is option B.