The following response simply adds some clarification to the response of Daniel L and is based on the same texts with some added nuances.
The proof (or support) that Aaron's staff is identical with Moses' staff is not found in the Numbers text, but is strongly suggested by the Exodus text. The following sequence of verses suggest the following three are the same:
The staff of God
Here are the verses and their implications
Exodus 4:2-4 introduces us to a staff that is converted to a snake and then back to a staff.
Exodus 4:17 explicitly identifies this staff as "the staff" with which Moses will perform miracles (before Pharaoh)."
Exodus 4:20 refers to the "staff of God" linking the description "staff of God" to the staff of Exodus 4. (Note how the next verse 21 connects this staff with the wonders that Moses is to do before Pharaoh consistent with Exodus 4:17.)
Exodus 7:15 explicitly describes the staff with which Moses performed miracles before Pharaoh as "the staff that was converted to a snake" providing consistency with all the above sources.
The following verse (actually paragraph) also links this staff to Aaron.
Exodus 7:8-13 relates the staff that turns into a tannin (snake or crocodile) before Pharaoh. In this paragraph, the staff is described as Aaron's staff.
It is very reasonable and plausible that this staff is not a separate staff but the staff spoken about in Exodus 4, and Exodus 7:15. In fact verses 8 and 9 read:
God spoke to Moses and Aaron. When Pharaoh speaks to you asking for a wonder(sign), then tell Aaron to take your staff and throw it before Pharaoh; it will become a tannin.
We can read the word "your" in these verses as referring to Moses. Aaron is commanded to take Moses' staff designated for miracles and to throw it before Pharaoh. Since Aaron performed on this occasion, the staff is also called Aaron's staff.
Returning to Numbers 17:18, there is no indication in these verses per se that the staff of Levi is in fact Aaron's staff which is in fact Moses' staff. However, it is called Aaron's staff and it is used for purpose of signs. It is very plausible that Moses took the staff designated for signs in Egypt and decided that that staff would be appropriate for a sign in the situation with Korach.