Definitely not with RAID5, as is the case with all other RAID5 solutions. The two are completely incompatible.
When I was moving my RAID5 running on Intel Matrix RAID to a Dell PERC controller, I purchased an extra single drive to migrate the data. Afterward that drive now functions as my backup for only critical data.
After countless stories and professional experience, any host-based RAID5 solution (e.g. AMD RAIDXpert, Intel Matrix RAID and nVidia RAID) have much weaker array integrity than pure software or hardware solution. I would suggest you look into either Windows' software RAID5 or *nix (FreeNAS recommended) running as VM for a free solution. Or if budget allows, a hardware solution which is normally ~$120-180 for used Dell PERC 5/i or 6/i found on fleaBay.
No you cannot move an onboard RAID array to different controller brands. It would if you used linux, of course. But with Windows you're out of luck. You need to backup all and re-create the array.
And perhaps you do not want to run RAID5 with onboard drivers. I'm assuming you have no proper backup and think the RAID5 is sufficiently protecting you from dataloss? If that is the case, you may want to setup two arrays instead and use one as backup; leaving you with less space but more reliability.