No, a SSD would not have a noticeable effect on your gaming performance. The biggest advantages of a SSD are how quickly it can load data - the OS benefitting most from this - and their improved durability (non-volatile flash memory vs. a mechanical disc with physical parts)
Here's a look at some fairly extensive testing that was done to answer this exact question.
As you can see, testing the same PC with a standard HDD vs. a SSD had absolutely no relevance on the performance of the games tested and in one case, the standard drive had higher minimum FPS than the SSD.
If you're having FPS issues, your GPU and RAM are what you need to look at upgrading, or possibly your CPU/Motherboard if your PC is more than a few years old.