Most of the observational studies of supernova (SN) explosions are limited to early phases (<a few years after the explosion) of extragalactic SNe and observations of SN remnants (>100 yr) in our Galaxy or very nearby galaxies. SNe at the epoch between these two, which we call the "transitional" phase, have not been explored in detail except for several extragalactic SNe including SN 1987A in the Large Magellanic Cloud. We present theoretical predictions for the infrared (IR) dust emissions by several mechanisms; emission from dust formed in the SN ejecta, light echo by circumstellar (CS) and interstellar (IS) dust, and emission from shocked CS dust. We search for IR emission from six core-collapse SNe at the transitional phase in the nearby galaxies NGC 1313, NGC 6946, and M101 by using the data taken with the AKARI satellite and Spitzer. Among six targets, we detect the emission from SN 1978K in NGC 1313. SN 1978K is associated with 1.3 x 10(-3) M-circle dot of silicate dust. We show that, among several mechanisms, the shocked CS dust is the most probable emission source to explain the IR emission observed for SN 1978K. IR emission from the other five objects is not detected. Our current observations are sensitive to IR luminosity of >10(38) erg s(-1), and the non-detection of SN 1962M excludes the existence of the shocked CS dust for a high gas mass-loss rate of similar to 10(-4) M-circle dot yr(-1). Observations of SNe at the transitional phase with future IR satellites will fill the gap of IR observations of SNe with the age of 10-100 yr, and give a new opportunity to study the CS and IS environments of the progenitor, and possibly dust formation in SNe.