Among two bodies of same mass, specific heat capacity but different coefficients of linear expansion which would require more heat for same change in temperature considering concepts of thermal expansion?
So long as the specific heat capacity of the two materials remain the same, it will take the same amount of energy to increase the temperature in both materials. That is because the heat required is defined through $Q = m C \Delta T$. By definition they will require the same amount of heat to change the temperature.
I understand your question as "doesn't some of the energy go into changing the materials properties?", and the answer is that if it does, this will be factored into the definition of $C$.
To me your question implies that you are asking about solids but even so the specific capacity will depend on not only the material but also the temperature and the expansion or otherwise of the material.
Because of this dependence of the specific heat capacity on a number of variables it is usual to quote the value of specific heat capacity for a material at a given temperature and then quote values when the material is not allowed to expand which is called the specific heat capacity at constant volume $C_V$ and when the material is allowed to expand at constant external pressure $C_P$.
The difference between these two so called principal specific heat capacities is significant for gases but very much less so for liquids and solids primarily because for gases their volume changes much more than liquids and solids for a given temperature change.
So if it is $C_V$ which you are making equal then that does not include the work which must be done (extra heat added) when a material expands.
In that case $C_P$ will be larger for the material which has the larger coefficient of thermal expansion.
On the other hand if it is $C_P$ that you have assumed to be equal then that already includes the heat input which has to be added to expand the material.
If you look at tables of specific heat capacities (e.g. Kaye and Laby) you will note that both $C_V$ and $C_P$ (or their ratio) are listed for gases whereas only $C_P$ is listed for liquids and solids because for most purposes when their temperature changes liquid and solids can be assumed not to change their volume by very much and the specific heat capacity is measured at constant pressure.