It won't quite be an upgrade. If you really want to upgrade, switch to the Ryzen processors, and get either of the Ryzen 5 processors, avoid Ryzen 3 for heavy gaming though. The best Ryzen processor for gaming would be a Ryzen 5 1600.
A G4560 wouldn't offer much over an i5 4440, but it does depend on what is the limiting factor in your current system. If you have a clock speed-heavy workload, like playing games at high refresh rate, then a G4560 might be a slight improvement, though an overclockable CPU would be more ideal for such uses.
However, most of the games and almost everything else as well nowadays is multi-threaded, so having more cores increases performance. The i5 you have right now is having 4 cores and 4 threads, while the G4560 has only 2 cores and 4 threads. So depending on what you use your computer for, the G4560 will actually be a downgrade in most cases, not an upgrade. If you have a multi-threaded workload, which you most likely do, then your upgrade options would be an i7(older or newer), or a Ryzen 5 or Ryzen 7 chip. If neither of those are in your budget, it's best you save up right now until you can afford a Ryzen 5(or Intel's upcoming Coffee Lake), because that i5 is still going to work for some time, in games at least.
But if you do have a clock speed-dependent workload, then a G4560 will offer slightly better performance. The G4560 has a base speed of 3.5 GHz, while the i5 only has a base of 3.10 GHz. So the additional 400 MHz(0.4 GHz~=400 MHz, right?) will help a little bit, not much though.