Understanding the impact of the astrophysical environment on Type Ia supernova (SN Ia) properties is crucial to minimize systematic uncertainties in cosmological analyses based on this probe. We investigate the dependence of the SN Ia SALT2.4 light-curve stretch on the distance from their nearest galaxy cluster to study the potential effect of the intracluster medium (ICM) environment on the intrinsic properties of SN Ia. We used the largest SN Ia sample to date and cross-matched it with existing X-ray, Sunyaev-Zel'dovich, and optical cluster catalogs in order to study the relation between the stretch and the distance to the nearest detected cluster from each SN Ia. We modeled the underlying stretch distribution with a Gaussian mixture with relative amplitudes that depended on redshift and clustercentric distance. We find that the fit quality of the stretch distribution improves significantly when we included the distance-dependant term in the model with a variation of the Akaike information criterion $ AIC = -10.2$. Because of the known correlation between galaxy age and distance from the cluster center, this supports previous evidence that the age of the stellar population is the underlying driver of the bimodal shape of the SN Ia stretch distribution. We further computed the evolution of the fraction of quenched galaxies as a function of distance with respect to cluster center from our best-fit model of the SNe Ia stretch distribution and compared it to previous results obtained from $H line measurements, optical broadband photometry, and simulations. We find our estimate to be compatible with these results. The results of this work indicate that SNe Ia searches at high redshift targeted toward clusters to maximize detection probability should be considered with caution as the stretch distribution of the detected sample would be strongly biased toward the old subpopulation of SNe Ia. Furthermore, the effect of the ICM environment on the SN Ia properties appears to be significant from the center of the clusters up to their splashback radius. This is compatible with previous works based on observations and simulations of a galaxy age gradient with respect to clustercentric distance in massive cluster halos. The next generation of large-area surveys will provide an increase of an order of magnitude in the size of SN Ia and cluster catalogs. This will enable us to analyze the impact of cluster mass on the intrinsic properties of SNe Ia and of the fraction of quenched galaxies in the outskirts of clusters in more detail, where direct measurements are challenging.