The number of lattice points in d-dimensional hyperbolic or elliptic shells {m : a<Q[m]<b}, which are restricted to rescaled and growing domains r,Omega , is approximated by the volume. An effective error bound of order o(r^{d-2}) for this approximation is proved based on Diophantine approximation properties of the quadratic form Q. These results allow to show effective variants of previous non-effective results in the quantitative Oppenheim problem and extend known effective results in dimension d ge 9 to dimension d ge 5. They apply to wide shells when b-a is growing with r and to positive definite forms Q. For indefinite forms they provide explicit bounds (depending on the signature or Diophantine properties of Q) for the size of non-zero integral points m in dimension dge 5 solving the Diophantine inequality |Q[m] |< varepsilon and provide error bounds comparable with those for positive forms up to powers of log r.