Hello Marta,
The best approach is probably to convert the .mov files to .mp4 (maybe consult this encoding guide) and replace them via Edit->Linked Files .
Alternatively you could install the VLC media player and see if the videos then work (well enough) in ELAN.
Finally you could try this: in the ELAN installation folder, in the app sub-folder, there is a file named elan.properties. Open this in a text editor, locate the line:
#JMMFPlayer.ForceForAllFileTypes=true
remove the leading # sign, save the file and (re-)launch ELAN. Maybe the .mov files then work in ELAN (by the JMMFPlayer, which is based on Microsoft Media Foundation).
-Han