What does the area defined by two different values for quantity under a supply curve represent? Assume that a and b (b is greater than a) indicate the quantity of supply on a linearly increasing supply curve. How can be the area formed by those two points under the supply curve interpreted? The interpretation should be made independent of a demand curve. That means no equilibrium price and producer surplus to account for. Could that be the difference between production costs for the given range of quantity?
Your intuition is right, but you are not accounting for fixed costs.

The supply curve is the part of the marginal cost curve above its intersection with the average variable cost curve. Marginal costs are defined as the derivative of the cost function (or the variable cost function as the derivative of fixed cost is zero).

To get the area you integrate. That is, you get back the variable cost.
So the area below the supply curve between $a$ and $b$ is the difference of the variable costs.