No no, different phones, same manufacturer. Full name of Nexus 5 is LG Google Nexus 5....Google had LG make it for them. I own a Nexus 5....you can hardly go wrong with them, superb display, quick and responsive, good camera, relatively inexpensive because Google use it as a platform for their Operating System, from which they make money. You get System updates first on Nexus devices.
The G2 I believe is regarded as overall a better device....especially noted for its long battery life.
Have a read of this: