You don't have to consider the value of the card at all. Ignore whether the card is a queen, king, or whatever. Just look at the suit.
The probability you draw 3 consecutive hearts is equal to the probability that the first card is a heart and the second card is a heart and the third card is a heart.
What's the chance the first card is a heart? Well there are 13 hearts out of 52 cards.
13/52 = 1/4
What's the chance the second card is a heart? Remember there are only 12 hearts left out of 51 cards:
What about the third card being a heart? (11 hearts out of 50 cards)
Multiply this out:
1/4 x 12/51 x 11/50 = 0.0129411765
Edit: Ah! I see your confusion with the word "consecutive". If that were the question, it would be interesting to solve. The ways to pick a set of 3 hearts would be 13 choose 3 = 13 x 12 x 11 / 3! = 286 ways. Assuming that ace was only high or low, there are only 11 ways to form a hand with consecutive hearts (A23, 234, 345, 456, 567, 678, 789, 8910, 910J, 10JQ, JQK). So your answer should be 11/286 times as small.
P(3 hearts that form a consecutive sequence of values)
= 13/52 x 12/51 x 11/50 x 11/286 = 0.000497737557
Approx. 1 in 2000.