Counterfactual explanations provide actionable changes to input feature values that would lead to a different model prediction. While previously studied for point prediction, their use in uncertainty-aware settings remains underexplored. In this paper, we introduce an approach for generating counterfactual explanations for conformal regression, where predictions are expressed as valid intervals. We define a set of different interval level properties that counterfactuals should satisfy and propose an optimization-based method for generating such counterfactuals. Beyond established evaluation metrics like distance and sparsity, we introduce two novel measures tailored to conformal prediction: plausibility, measured using the conformal predictive distribution, and difficulty, measured using the variability of predictions around a counterfactual. Experiments across five tabular datasets demonstrate that counterfactuals satisfying the selected property can be generated while achieving low distance, sparsity, difficulty, and high plausibility. This opens new directions for uncertainty aware explanations for conformal regression.