Answer
No.
Work Step by Step
For example, consider a red-hot iron nail thrown into a swimming pool. The iron's change in temperature is much larger than the water's.
In general, if heat flows from one to the other, the amount of temperature change will be the same only if the product of mass m and specific heat capacity c is the same. This can be seem from the equation $Q = cm\Delta T$.
One example of when this might happen is when both objects are the same mass, and composed of the same material.