Pyroglutamate can form from the cyclization of either glutamic acid or glutamine. In either case, the alpha amino group of the amino acid becomes linked to the delta position of the amino acid through an amide bond (because it is cyclic, it is sometimes referred to as a lactam). Note that in forming the ring, the alpha amino group is lost. That means that the peptide or protein cannot be extended because there is no amine available to form the next peptide bond. So, no, pyroglutamate cannot be internal to a peptide chain.
But that does not mean that you cannot find the pyroglutamate structure elsewhere. It is a common structure in drug design. Do a substructure search in PubChem or Chemicalize and you will find dozens of molecules that contain a pyroglutamate structure. Many of these are parts of chemicaal libraries used in teh search for new drugs. It is possible that a pyroglutamate structure could be found in nature but if so, it will not be part of a peptide chain unless it is at the N-terminus. (I could easily imagine synthesizing a peptide that has a pyroglutamate group attached to the epsilon amine group of an internal lysine but that would not be through a peptide bond ( a special type of amide). It would be a regular amide bond. I have never seen any examples of this but that does not mean that they do not exist.)
Dear Luigi, just to correct that during the mucleophilic attack of N from the primary (alpha) amino group on C atom of epsilon amide group, it is the epsilon NH2 group that leaves the reaction as ammonia, and not the primary amine as specified above.