In this work we construct a unitary operator acting between Spin ^c quantizations of compact integral symplectic manifolds which are symplectically cobordant. The construction is based on a recent proof of the cobordism invariance of the index of Dirac operators, due to L. Nicolaescu. We prove that, away from a small set on the boundary manifolds, this quantized cobordism operator is an Hermite-Fourier integral operator in the sense of L. Boutet de Monvel and V. Guillemin. The proof consists of two parts. In the first we show that the Spin ^c Szego&huml; projector, II, which defines the quantization, is itself an Hermite-Fourier integral operator just as in the well studied case of Kahler quantization. In the second part we explicitly construct the fundamental solution, K, of a boundary value problem for a transversally elliptic Dirac operator on a principal circle bundle over the cobordism. This construction is carried out within the category of Hermite-Fourier distributions and relies on their symbolic properties. For example, the transport equations are evolutions equations on <blkbd>R2n,</blkbd> in contrast with the case of Lagrangian distributions where they are ordinary differential equations. The microlocal structure of the quantized cobordism operator is the same as that of the composition P&j0;g&j0;K, where gamma is the boundary restriction operator. From the composition theorems for Hermite-Fourier integral operators it follows that P&j0;g&j0;K is an Hermite-FIO. In the last part of this work we present examples where our construction applies. Advisors/Committee Members: Uribe, Alejandro (advisor).