Although Type Ia supernovae (SNe Ia) are very useful in many astrophysical fields, their exact nature is still unclear, e.g., the progenitor and explosion models. The high-velocity features (HVFs) in optical spectra of SNe Ia could provide some meaningful information to constrain the nature of SNe Ia. Here, I show strong evidence that the SNe Ia with a strong Ca ii infrared triple (Ca ii IR3) HVF around maximum brightness are associated with a relatively younger population than those with a weak Ca ii IR3 HVF, e.g., the SNe Ia with a strong Ca ii IR3 HVF tend to occur in a late-type galaxy or in an early-type galaxy with significant star formation. In addition, using pixel statistics, I find that the SNe Ia with a strong maximum-light Ca ii IR3 HVF show a higher degree of association with the star formation index, e.g., H? or near-UV emission, than those with a weak Ca ii IR3 HVF. Moreover, I find that the strength of the Ca ii IR3 HVF is linearly dependent on the difference of the absorption-weighted velocities between the Ca ii IR3 and Si ii 635.5 nm absorption lines, which then is a good index for diagnosing whether there is a high-velocity component in the Ca ii IR3 absorption feature in the spectra of SNe Ia. I finally discuss the origin of the HVFs and the constraints from our discoveries on the progenitor model of SNe Ia.