We know Earth pulls every object by the same acceleration $g$ (say).
force exerted by earth $F=m\cdot g$ (with $m$ the mass of the body), so, if $g$ is constant then $F$ is directly proportional to $m$. So what will the force exerted by earth be if the body is a quaser or a black hole?
Yes, the force indeed is proportional to the mass. (Since $F_g=\frac{Gm_1m_2}{r^2}$) ( $m_1, m_2$ are masses of the bodies.)
But this doesn't really matter, because say, for $m_1$, $$m_1\vec a=\frac{Gm_1m_2}{r^2}$$ If you notice, the acceleration of the body is independent of its mass, like you mentioned.
Why do you think this will be any different for a blackhole or a quasar? (Ignoring GR, of course.)
The force you ask for is simply, $$\frac{Gm_\text{earth}m_\text{blackhole/quasar}}{r^2}$$ ($r$, being the distance between their center of mass(s).
