Spectral properties of QPSK modulation
The spectrum is that of the base pulse used in the modulation, so in this case a rectangular pulse. A single rectangular pulse in time, as given by the Fourier Transform, is a Sinc in Frequency with the first nulls spaced at $1/T$ away from the center of the Sinc, where $T$ refers to the duration of the pulse in time. (Just as that shown by the OP). If the pulse is repeated at a consistent rate, then then only spectral components would be weighted impulses spaced by the repetition rate, and the weights would be the same Sinc function shape shown. Thus a 50% duty cycle square wave would have its even harmonics in the nulls of the Sinc function shown, resulting in the well known odd-harmonic spectrum of impulses. If the data is random, such that a patter never repeats, then the spectrum will be continuous as shown here.