Yes, you can do this easily with QB Pro desktop Enhanced payroll. All you do is create two different sets of pay types: For example:

RegWA

RegID

OTWA

OTID.

Enter the hours for each pay type on pay day

You will need to set the lived in State as WA and worked in State as ID to also pull in the Idaho state taxes

You will probably need to over-ride the taxes calculated by QB because it  may assume all pay is in the Idaho tax jurisdiction. (I used an Excel spreadsheet.) All pay from both states will be subject to WA Employment Security and PFML. With the lived in state set to WA it should calculate those correctly. I processed payroll with EEs going back and forth between WA and OR so I learned this process very well.