Gaussian outputs dipole moment and quadrupole moment by default. For other programs, some of them output dipole and quadrupole moments while others do not. You can have a try, if the moments are not printed by default, you may search program manual to try to find relevant keyword.
By the way, subfunction 5 of main function 300 of Multiwfn (http://sobereva.com/multiwfn) is able to calculate dipole, quadrupole and octopole moments and output them in a very detailed way. If the quantum chemistry program you used is able to yield a wavefunction file supported by Multiwfn (wfn/wfx/mwfn/fch/molden/gms..., most mainstream programs support at least one of them), then you can easily use this function to obtain the moments. See Section 3.300.5 of Multiwfn manual for detailed introduction and Section 4.300.5 for example.
As Tian said, Gaussian prints the dipole moment and quadrupole moment in the output files by default (also the octapole and hexadecapole moments). This is the preferred software we use in our group, but we also use other options that work in a similar way such as TURBOMOLE or ORCA (free software).
You might need some time to get familiar with the input files of any of the programs mentioned above, since you need to take into account quite a few things (how to input functional, basis set, solvation models, charge, multiplicity, coordinates, etc).