This is not really a very fair question, as it depends how you define a major battle. There is no consensus on the Internet – in fact it's not all that easy to find a nomination.
When I googled "american revolutionary war first major battle" (using the American name for the war as I thought it would be more fruitful), Lexington and Concord featured heavily - Google seeming to ignore the word 'major'.
Keen to see things from a British point of view, I then googled "american war of independence first major battle". Some way down the list, I did find a link to a page on the British Library's website, which does describe Bunker Hill as "The first major battle of the War of Independence."
To summarise, I think it is probably fair to describe Bunker Hill as the first major battle of the War of Independence; but I'm not convinced that this is definitive enough to make it a fair quiz question, without some clarification.
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