We have theoretically studied electron velocity fluctuations and the resulting Johnson (thermal) noise in a free-standing GaAs quantum wire at low and intermediate driving electric fields. One-dimensional confinements of electrons and phonons have been taken into account. Acoustic phonon confinement introduces infrared frequency peaks in the noise power spectrum which are an unmistakable signature of phonon confinement and provide an experimental `handle' to use in assessing the importance of such confinement. Phonon confinement also suppresses the dc component of the noise spectral density (and the hot-carrier diffusivity) by several orders of magnitude. When a transverse magnetic field is applied to the quantum wire, it introduces three remarkable features: (i) it reduces the temporal decay rate of the velocity autocorrelation function and increases the dc component of diffusivity, (ii) it promotes prolonged and persistent oscillations in the velocity autocorrelation function which is indicative of a long memory of the electron ensemble, and finally (iii) it red-shifts the peaks in the noise power spectrum by increasing the length of an electron's trajectory in momentum space between two successive phonon scattering events.