Yes, I believe that the heatsink-fan unit that comes with the Pentium G4560 is rated for 65W, so should handle this Core i7-7700. If you are seeing processor temperatures above, say, 90 degrees, you might want to consider getting a better cooler.
How are you obtaining this Core i7-7700? If you are purchasing it used, then it has no warranty. Only the original owner has a warranty. This warranty becomes null and void when the processor is passed on to a second party. If you are purchasing it (supposedly) new as a tray processor (i.e. not boxed with a heatsink-fan unit of its own), remember that this processor is (a) not new and (b) it does not come with a warranty. Tray processors are intended to be purchased by OEMs and used in their system-level products; they are not intended to be sold to end users (nor anyone else making a grey market purchase). These OEMs will have a separate warranty agreement with Intel. Thus, no warranty entitlement comes with the processors themselves. Buyer beware! There's a good reason why these processors seem cheaper (but really aren't, as you would find out if you had to approach Intel regarding replacement).
...S
