No, controllers block access, not drives. There will be a difference in firmware between the non-Dell branded (generic/retail) Seagate and the Dell-branded Seagate, so there may be issues ranging from small to major, but if the enclosure/controller doesn't block non-certified drives, then at least you'll be free to test it, but if you run into problems, that's where I would start my troubleshooting.
I have a bunch of Dell-branded SAS drives that were part of old MD1120 3Gbps SAS enclosure.  Based on the labeling on the drives, they appear to be re-branded Seagate Savvio 10K.3 6Gbps SAS drives.  Can I use these drives in a non-Dell server/enclosure with a 6Gbs SAS backplane?  I just want to make sure the Dell drive will work with a non-Dell controller.  The enclosure and drives are out of warranty, so I'm not worried about technical support from Dell. I've seen lots of people asking about using non-Dell drives with Dell enclosures, but I want to do the reverse.
Dell drives in a non-Dell enclosure?