Perhaps there are some bugs in the high precision computation or misuse of data types.
My platform is MacOS High Sierra 10.13.2.

This is an integer overflow, which is intentional (docs).
Note that you can get the correct answer as follows:
julia> BigInt(1141241515151)*BigInt(124124124124124)
141655603482206064550602724
Note that in Matlab 1141241515151 and 124124124124124 are floating-point values, not integers, which lose precision instead of overflowing. You can get the Matlab-like behavior this way:
julia> 1141241515151.0*124124124124124.0
1.4165560348220606e26
That's really a cool design! Thank you!
Most helpful comment
Note that in Matlab
1141241515151and124124124124124are floating-point values, not integers, which lose precision instead of overflowing. You can get the Matlab-like behavior this way: