Yes, the Stefan Boltzmann law will tell you how much heat a blackbody will absorb from another blackbody.
Your equation is giving the net heat flowing from body 1 to body 2.
To break it apart a little and relate this to absorption.
Body 1 is emitting $e A \sigma T_1^4$ to body 2 and absorbing $e A \sigma T_2^4$ from body 2. Hence the net heat flow body 1 emits is the difference between these fluxes $e A \sigma T_1^4 - e A \sigma T_2^4 = e A \sigma \left( T_1^4 - T_2^4 \right)$ which is the equation you stated.
Regarding the absorption aspect of you question.
The main axiom regarding a blackbody is that it is a perfect absorber. So any radiation falling on it is absorbed and contributes to raising the temperature of the blackbody.
In your example, you are considering a blackbody with an emissivity/absorptivity. This is the factor of incident radiation which is absorbed, via Kirchoff’s law, it also reduces the emission rate of the body. Sometimes a non-ideal blackbody is called a grey body.