Peano Arithmetic suffices to prove Godel's Theorem, and Peano Arithmetic is consistent, so yes.
If you don't believe Peano Arithmetic is consistent, then the question becomes: What do you believe is consistent? As you can see in the accepted answer to this question, you can prove Godel's Theorem in Primitive Recursive Arithmetic, or even a bit less. If you doubt that PRA is consistent, you'll be forced to doubt so much that any skepticism you might have about Godel's Theorem is probably among the least of your worries.
