probate procedure after mother dies in WA state, dealing with quit claim deed

You likely should review your facts and options with an attorney.

It is not clear from your post whether you are the grantee of the quit claim deed. If someone else is the grantee, you may have other problems.

One issue you would want to discuss with the attorney is taxes. Depending on the value of the house and your mother's cost basis in the house, there may be quite a bit of difference in the amount of various taxes.

Documents that convey interest in real property must contain statutory elements and statutory procedures must be followed. The quitclaim deed may not be valid.

An unrecorded Quitclaim Deed kept in the possession of its Grantor cannot be used to transfer property after the Grantor's death, as a violation of the Statute of Wills (enacted in England in 1540). According to the Statute of Wills, if an owner or property desires to transfer at his or her death property titled solely in his or her name, such must be made by Will, with all of its formalties (for example, a writing signed by its maker and witnessed by two other persons; it does not need to be notarized). A Deed is not made with the same formalities as a Will (for example, a writing signed by its maker but not witnessed, only notarized) and therefore is ineffective to transfer property resulting from the death of its Grantor. The only exception to this rule in my awareness is if the Grantor surrenders possession & control of the Deed to an independent third party and instructs the third party to record the Deed upon the Grantor's death.

Answer: The Will.