Converts glutamine and N-terminal glutamyl residues in peptides to 5-oxoproline and 5-oxoproline residues. Not involved in the major pathway for 5-oxoproline production. N-terminal L-glutaminyl-[peptide] = N-terminal 5-oxo-L-prolyl-[peptide] + NH4(+) Requires an unprotonated substrate N-terminus and a protonated basic group of the enzyme for binding and conversion of the glutaminyl substrates. Optimum pH is mildly basic for conversion of the glutaminyl substrates and shifted to acidic for cyclization of glutamic acid. Glycosylated. No effect on 5-oxoproline production. Has a low metal affinity, in contrast to the mammalian glutaminyl cyclases that contain one zinc ion per molecule. Belongs to the plant glutaminyl-peptide cyclotransferase family.