Heat flows between two bodies in contact until the system attains thermal equilibrium. What happens if the bodies in contact having different specific heat capacity and same temperature. Does the flow of heat occurs until the heat energy between the bodies become equal since the amount of heat need to rais the temperature is different?
May be an analogy will help. If two tanks containing water are connected by a pipe, water flows from the tank with higher level to that with lower level of water. Temperature is analogous to water level. Heat capacity is analogous to floor area of the tank. The tank with a larger floor area can hold greater amount of water (for a given water level), but that has nothing to with determining whether and in which direction water will flow.
Simply a No.
There are so many ways to reason this. Suppose it's possible and the heat is transferred from object A, to object B, with initial temperature $T_A = T_B = T_o$. With heat transfer alone, $T_A$ decreases and $T_B$ increases.
Since you have two sources with different temperatures, you would be able to build an engine out of it. This, violates the second law of thermal dynamics.
Or simply, when they are of different temperatures, heat transfer will happen until they again reach the same temperature.