I contacted the author of this package (Anton Antonov) and not only did he reply in very short notice, he was kind to offer a solution, to which I post the code below.
If you are interested in learning how the QuantileRegressionFit function works, check out Anton's GitHub.
Cheers,
Thad
In[66]:= data = Array[Prime, 300];
data = Transpose[{Range[Length[data]], data}];
In[68]:= funcs = {1, x, Log[x]};
In[70]:= qres = QuantileRegressionFit[data, funcs, x, {0.5, 0.7}]
Out[70]= {3.16622*10^-10 + 6.14692 x + 1.9274*10^-10 Log[x],
4.9249*10^-12 + 6.34137 x + 8.47339*10^-13 Log[x]}
In[71]:= MapThread[Join, {Map[Cases[#, _?NumberQ] &,
qres] /. {} -> {0},
Outer[Coefficient, qres, DeleteCases[funcs, _?NumberQ]]}]
Out[71]= {{3.16622*10^-10, 6.14692, 1.9274*10^-10}, {4.9249*10^-12,
6.34137, 8.47339*10^-13}}