Based on the information provided on the question it's not possible to know of the router is performing NAT or not. However a simple test can be used to determine if NAT is in use:
If you can successfully ping PC B from PC A and vise-versa using each PC's actual IP address, then the router is not performing NAT. (This assumes there are no firewalls blocking ping, so you may need to disable any, including on the PCs.)
Conversely, if NAT is in use, the PC "behind" the NAT will be able to ping the other computer, but it won't work the other way around. For example, of PC A is behind a router doing NAT, it would be able to ping B, but B would not be able to ping A (using A's actual IP address). This is because NAT translates (i.e. masquerades) PC A's IP address to make it appear as though it has an IP address on the same subnet as PC B. This is useful when you have a lot of devices on subnet A, each with their own IP address, but you only have one IP address on subnet B (which could be the Internet). NAT let's all those devices communicate with the rest of the network/Internet through a single IP address. This is used extensively to mitigate the impact of IP v4 exhaustion.
If NAT is not in use, then the router simply does what all routers do...it sends packets from PC A destined to PC B out the correct interface so they reach their destination. Most routers have multiple interfaces connected to a variety of networks, such as may be the case in an organization where multiple buildings each have their own network connected by a central router.
Most "routers" that connect two networks, specifically a LAN to the Internet, are not actual routers in the strict sense, but are Residential Gateways.
ip route add ...
on the root account, or whatever configuration files it uses.