how are your four monopole antennas feeded? Usually you have an electrically conducting plane with the monopole antenna aligned perpendicular to this plane, and a coax cable feeding the antenna from behind the plane. But combining four antennas in such a way that they form a square (if I understand correctly) would cause the ground planes to intersect each other resp. each antenna were aligned parallel to the ground plane belonging to another antenna. In other words, the four planes would form some kind of a rectangular waveguide, with the antennas inside the guide.
I would say "NO". It doesn't. orthogonal dipoles function as a circular polarization antenna if you just excite them with 90 degree phase difference. Otherwise you will have two linear polarized antenna. There are many solutions to apply the phase difference between ports. A delay line with a quarter wavelength is extensively reported. Take a look at following paper:
Two orthogonal dipoles with a phase difference of 90 degrees can produce a circularly polarized wave. Without a proper feeding mechanism and a phase difference, the two antennas shall generate linearly polarized waves.