Simple Javascript Score Object

Here's how I would do this...

First, I'd simplify the scores array so that access is trivial and easy to know exactly what I'm dealing with, and when. I've also got a couple of variables that will be used to show the final score.

var rounds = 1;
var maxRounds = 4;

var names = ["Spring", "Summer", "Autumn", "Winter"];
var winners = [];

var scores = [];
names.forEach(function(current) {
  scores.push({ name: current, score: 0 });
});

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With this change, the questionCalc function simplifies a bit as well...

// *** IMPORTANT NOTE: 
// You need to ensure that 'number' is zero-based and smaller than the scores (or names) array length.
function questionCalc(number) { 
	if(rounds === 1) {
		winners = [];
		scores.forEach(function(current) {
			current.score = 0;
		});
	}
	
	scores[number].score++;

	var results = "RUNNING RESULTS ";
	scores.forEach(function(current) {
		results += current.name + ": " + current.score + " ";
	});
	console.log(results.trim());
	
	if(rounds === maxRounds) {
		determineWinner();
		rounds = 1;
	} else {
		rounds++;
  }
}

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Finally, a simple descending sort on the scores array makes determining the winner very easy.

function determineWinner() {
	scores.sort(function(a, b) {
		return b.score - a.score;
	});

	scores.forEach(function(current) {
		if (current.score === scores[0].score) {
			winners.push(current);
		}
	});

	var multipleWinners = winners.length > 1;
	console.log((multipleWinners ? "multiple" : "1") + " winner" + (multipleWinners ? "s" : "") + "... ");
	winners.forEach(function(current) {
		console.log(current.name + "! (score: " + current.score  + ")");
	});
}

// use a button or something to run the function
$("addPoint").click(function(){
	var recipient = Math.floor((Math.random() * maxRounds) + 1);
	questionCalc(recipient);
});

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Here's a working Fiddle (with some minor updates and obligatory embellishments, of course) that illustrates the action. I noticed that JsFiddle's auto-save functionality interrupts the action sometimes... so it may be useful to pause a second or two between each button click (you'll see what I mean).

Also, I'd normally use the traditional for (var i = 0; i < {array}.length; < i++) construct, but my Fiddle started acting nuts for no reason at all... so I went with what worked (because I was tried and didn't feel like fighting someone else's code).