It should be compatible as 390 is validated to run many HD ##50 and ##70 gpu such as 6770 there is no reason to suspect 6750 is an outlier.  try replace cmos battery with a new battery and clear cmos settings via motherboard RTCRST jumper.  It is probably corrupted 390 bios settings that is blocking 6750. a popular list of gpu validated by users in 390  
Hi, everyone! Hope you're all having a nice life!  I recently got an optiplex 390 MT, and all good. I have an old xfx radeon hd 6750 and was hoping I could use it on this pc, however, it doesn't work; I put the GPU in the PCI-e slot, rebooted, nothing happened, nada. The card worked last time I used it. Is this GPU compatible with optiplex 390?  My PC specs: Processors: 4 × Intel® Core™ i3-2120 CPU @ 3.30GHzMemory: 7.7 GiB of RAMGraphics Processor: Mesa Intel® HD Graphics 2000Manufacturer: Dell Inc.Product Name: OptiPlex 390System Version: 01 BIOS: A14. Any help is much appreciated. Thanks in advance for your answers.
Radeon hd 6750 on dell optiplex 390