PCIe design

There has to be a root complex (RC) in any PCIe system. If the FPGA and DSP core are the only two devices that are on the PCIe bus, then one of them must be the RC, yes.

You haven't stated what your DSP board is. However it is likely that it doesn't have the capability to be the RC - devices are either an endpoint (EP) or the RC, they cannot (?) be both. If your DSP board doesn't have the capability to be an RC, then the FPGA must be configured to perform that task.

The Altera PCIe Hard-IP cores support RC mode so you should be able to set it up. Though be under no illusion that this is a simple task - PCIe is complicated and setting it up requires some effort. There are sample designs that can help, but even those have a steep learning curve.

If you don't want to set up an RC, you could plug both the FPGA and DSP cards into a computer. The computer has an RC built in to the CPU or chipset, which means that both of your devices can then be end points. The computer will do the work of configuring the devices and the PCIe bus, and then once set up the FPGA and DSP can talk to each other autonomously.