No. It always occurs, even if the product of the reaction is less stable than the reactant - energy is always released in the formation of bonds, regardless of whether the overall reaction is endergonic or exergonic. That's why bonds form in the first place. I think you're missing the intent of the original question. Several of the posters in this thread have remarked that energy is released in a reaction when the products are more stable than the reactants, which is true, but not what the OP was asking.
Whether an overall reaction releases or consumes energy is irrelevant to the question of whether bond breakage requires an input of energy or not. You seem to be focusing on the reaction energy, which is the finished product, while skipping over the parts in between. Let me try to simplify and maybe what I'm trying to say will make more sense:
- The cleavage of any covalent bond always requires an input of energy
- The formation of any covalent bond always releases energy
- One can estimate the change in enthalpy for a reaction by subtracting the sum of the enthalpy changes for all the bonds formed from the sum of the enthalpy changes for all the bonds broken
If the change in enthalpy is negative, then the reaction is exothermic and if it is positive, than it is endothermic. Regardless, bond cleavage requires energy and bond formation
always releases energy. The relative amounts for both quantities ultimately determines whether the overall reaction releases or requires energy.
Whether a reaction is favorable or not requires a knowledge of the entropy change of the reaction, which can be determined in a similar way. Once the changes in enthalpy and entropy are known, you can determine favorability by calculating the change in the Gibbs free energy of the reaction. It's entirely possible to have an exothermic reaction which is not favorable at normal temperatures.