Near-infrared images of elliptical galaxies with morphological signatures of recent merger activity have been obtained. If this merger activity stimulated significant and distributed star formation within the region defined by R < 1.5R_e, then JHK photometry in this region should detect the presence of an intermediate-age stellar population associated with this event. Our observations, however, show that these galaxies occupy the same locus of points in the J - H versus H - K plane as elliptical galaxies with no signs of recent merger activity. This strongly constrains the global fractional amount of intermediate-age (1-3 Gyr) stellar mass to be no more than 10%-15%, and in most cases the data are consistent with all the stellar mass being old (T > 10 Gyr) and metal-rich ([Fe/H] >= -0.3). We argue that any recent merger activity was, therefore, not accompanied by a significant episode of distributed star formation within the region defined by R < 1.5R_e and that the bluer UBV colors often observed in this region for some of these galaxies are due to the accretion of lower mass companions or relative abundance differences. The possibility of a strongly centralized starburst resulting from the merger, instead of a more distributed star-forming episode, is fully discussed in Paper II of this series, in which similar tight constraints on this activity will be discussed.