Even if quad precision is not directly supported by most CPUs, many Compilers (GNU, Intel) support them. Also some software packages allow to compile with quad precision, e.g. PETSc. But is there really a need for this? I asked some days ago about Krylov subspace method in finite arithmetic, see Krylov subspace iterative methods in floating point arithmetic. I read the mentioned papers, and it seems to me that these methods are well understood in floating point arithmetic. Does iterative solvers benefit from using higher precision if the matrices are well conditioned? I.e., does the iteration counter goes down with higher precision? Are there, may be, some studies on this topic? Are there problems in scientific computing, where double precision is not enough to formulate and discretize them?
I think this is actually a duplicate of this question: Higher precision floating-point arithmetic in numerical PDE
As I stated on that question, quad precision is certainly not widely used in scientific computing. Moreover, I do not believe that quad precision is actually necessary in any significant number of applications.
This does not answer your question, but there are people - interested in the reproducibility and reliability of large scale simulations, e.g., climate simulation - who investigate implementational details of floating-point algorithms in double precision.
Behavioural differences have been observed, e.g., when the scalar products in the CGM are computed not by naive summation, but by more sophisticated algorithm as compensated summation or summation in order of ascending magnitude. This is particular relevant for parallel summation on many cores, because the summation (say, in MPI) is usually black-boxed.
A search like ["conjugate gradient" "compensated summation"] in a well-known search engine for scientific papers will give a few (!) publications on that. As far as I remember, the iteration count might vary slightly, and might even increase when improving the numerical accuracy. It is probably less relevant on your local quad-core work station.
High precision is often used to produce coefficients accurate to double precision. For example, if your trying to compute a Butcher tableau you might do it in quad, so that every bit is correct in the double precision code. Similar things occur when computing Gauss-Kronrod quadrature nodes, or filter coefficients in DSP.
This is more for library writers than for application developers.