What is the air pressure in the heliosphere (Sun's atmosphere)? Specifically, I am wondering what the pressure is at a distance of around 1 AU. Does it it decrease quadratically with distance?
I am going to assume that what you mean here is "what is the pressure" in the solar wind? There is no "air"!

The solar wind is pretty complex, consisting of a "fast wind" observed predominantly at high solar latitudes and a slower more variable wind a low latitudes. Both components essentially consist of an expanding stream of protons, electrons plus a small fraction of helium nuclei.

I think all the information you want is contained in this paper by Ebert et al. (2009), but I will attempt a broad-brush summary.

The fast wind has a typical proton number density of $n_p=3$ cm$^{-3}$, a temperature of $T=2\times10^{5}$ K and a speed of $V=700$ km/s. Note that "temperature" is a slippery concept here since the velocity distribution of the particles is non-Maxwellian. However, ignoring this and assuming the protons are matched by numbers of electrons, then the "thermal pressure" $P_{th} = (n_p + n_e)kT \simeq 10^{-11}$ Pa. However, this is negligible compared with the dynamic pressure $P_d = \rho V^2$, where $\rho$ is the mass density given roughly by $\rho = n_p m_p$. Thus $P_d \simeq 2\times 10^{-9}$ Pa (i.e. about $2 \times 10^{-14}$ bar).

The "slow wind" is quite variable, but mean values could be $n_p = 6$ cm$^{-3}$, $T=8\times 10^{4}$ K and $V=400$ km/s. Here again, the thermal pressure $P_{th} \simeq 10^{-11}$ Pa is dwarfed by the dynamic pressure $P_d = 1.6 \times 10^{-9}$ Pa.

The thermal pressure component falls as distance from the Sun cubed, ie. as $P_{th} \propto R^{-3}$. This is because for mass conservation, then a sherically expanding wind must have a density that falls as $R^{-2}$, but it is experimentally found that the proton temperature also falls as $\simeq R^{-1}$.

The dynamic pressure does fall almost as $R^{-2}$. This is because there is very little radius dependence for the proton velocity (i.e. the protons do not slow down, right out to the heliopause - as measured for instance by Voyager 2).