Does Cyclone IV GX support PCIe Gen2?

If not mistaken, the CIV GX do not support 5Gbps transceiver data rate. You can double check the datasheet to verify.

CIV GX max data rate is 3.125G. So it would not be able to support PCIe Gen2 data rate.

You might need to select different device families ie SIVGX, CV GX to supprot Gen2. You can check for the protocol supports in the device handbook transceiver chapter.