Let R be a standard graded Gorenstein algebra over a field presented by quadrics. Conca-Rossi-Valla showed that such a ring is Koszul if reg (R)<= 2 or if reg(R)= 3 and codim(R)<= 4, and asked if this is true for reg(R)= 3 in general. We give a negative answer to their question by finding suitable conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization of R with the (twisted) canonical module is a non-Koszul quadratic Gorenstein ring.