How can I compute molar concentration if I am given parts per trillion?
I have the feeling, that this is a superbly stupid question. And I also have the feeling that it is quite possible, that my answer will be wrong.
First of all the parts-per-notation should be avoided as it is not compliant with SI and highly ambiguous.
Secondly, it generally may only refer to a unitless number - a molar concentration has a unit ($\mathrm{mol/L}$).

The here used unit may refer to a couple of things. In general its meaning should most likely refer to $\mathrm{PPTV = 10^{-12}L/L}$. Sometimes the use of $\mathrm{PPTV = 10^{-12}g/L}$

From here you can probably make your best guesses, determining how many moles of compound do you have:
\begin{align}
pV &amp;=n\mathcal{R}T\\
n  &amp;= \frac{pV}{\mathcal{R}T}\\
n  &amp;= \frac{101325~\mathrm{Pa}\cdot 261\cdot10^{-12}~\mathrm{L}}{8.314~\mathrm{\frac{J}{mol\cdot K}}283.15~\mathrm{K}}\\
n  &amp;= 1.12\cdot10^{-8}~\mathrm{mol} &amp;\implies c&amp;=  1.12\cdot10^{-8}~\mathrm{\frac{mol}{L}}
\end{align}

If you have the mass given, then you need to know which gas it is, as $n=\frac{m}{M}$.

Another possible use could be the particles by volume notation, so it could also refer to $\mathrm{PPTV = 10^{-12}/m^3}$. (Apparently this is somewhat common in ecological air measurements.) Here you simply have to use $n=\frac{N}{\mathcal{N}_\mathrm{A}}$ to figure out the concentration in $\mathrm{mol/L}$.