Can a $\gamma$-ray photon give some of his energy to an atom and accelerate it? I know gamma-ray photon can only give its momentum energy to the electrons of an atom.
My question is: Can a photon give some of its momentum to the atom (including its nucleus) to give it heat or speed?
If yes, can you tell me how much energy can it give?
 A: It is not very clear to me if you are asking about energy or momentum. You should also ask about a specific interaction process as there are many, this is required especially to answer your last, quantitative, question.
However, generally speaking, a $\gamma$ photon cannot give some of its energy to anything else: it is all or nothing. Even in the Compton scattering, in which you get a less energetic photon, the initial photon is destroyed. The momentum must be conserved as well, so yes: when a photon hits another particle this is accelerated, you can even generate some measurable pressure with a very intense radiation!
A: Not merely can it transfer its momentum as well as its energy when it interacts, but it must.
If the target atom is in a fluid context (liquid, gas, plasma), then that energy and momentum must end up in the target or some other reaction products(s).
In a solid context the Mössbauer effect can be an issue, allowing the transfer of that momentum to a much more massive collection of mass resulting in lower recoil.
It is also worth noting that this requirement to transfer both energy and momentum is the reason a photon can not be absorbed on a isolated point target in free-space:
$$ \gamma + e \to e \tag{forbidden} \,,$$
though the same reaction is possible in the field of a massive partner
$$ \gamma + e + A \to e + A \tag{allowed}$$
or if the target is a compound object with excited states
$$ \gamma + A \to A^* \tag{allowed} \,.$$
